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There's been this looming ghost of martial classes and spellcasters being compared to eachother, and the repeated quiet sentiments that the latter are just... sub-par. So, in a vain attempt to end the doubt for at least some people, I've decided to hop right into the ring and lay out in detail how exactly this part of the game has been carefully designed, how the math behind it ticks, what the little quirks and asterisks are, and why spellcasters are ultimately powerhouses and in an overall fantastic spot right now.
On Martials
You always need two points to make a comparison, so we've got to talk about martial characters for a moment. Martial characters have some defining features throughout:

• They don't have many abilities, but those are very strong abilities, and they can be used infinitely. Fighter has the classic +2 accuracy, and starts with Attack of Opportunity. Inventor has a cool invention with special unstable abilities and consistent damage boosts.
• They do a lot of damage. Like, a lot lot. Most of their abilities enhance the already high output of Strikes with magical martial weapons, and those that don't (such as Champion's Reaction, Monk's high speed and action economy or Outwit Ranger) offer very strong other benefits.
• Their defenses are superior.
• Getting additional abilities (f.e. Athletics maneuvers, Demoralize, Medic Dedication) for them is pricey in terms of feats, ability boosts, and/or skill increases.

They are specialists at their things - they do a low amount of things really well, and going beyond that is tied to opportunity costs. That low amount of things also includes one of the most important things in a game like this: Single target damage. They can focus fire to knock out weak targets quickly and very effectively apply themselves to kill enemies, which is how fights are won.
So if martials win fights... what is the point of spellcasters?
On Spell effects
Well, what is it spellcasters get? Various degrees of average-to-poor defenses, cantrips, and limited spell slots to cast things with. Let's pick four popular good spells to keep an eye on thoughout this analysis:

• Fireball. Lv3 spell, 6d6 damage in a 20ft burst on a basic save, with an extremely high range.
• Heal. Lv1 spell, 1d8+8 healing at 30ft range or other more niche but occasionally more useful applications.
• Slow. Lv3 spell, slowed 1 at 30ft range. Lasts a minute on failed save, or a round on passed save.
• Air Walk. Lv4 spell, allows you to walk through the air for 5 minutes.

Right off the bat, it can be noted that none of these effects are things martial characters can achieve, or at the very least not at those levels.

• Fireball's 6d6 is quite high by even by the standards of a full turn of single-target ranged martial attacks, and has far more range, area and damage than any analogues like Inventor's Explode. It also covers a HUGE area.
• Heal allows for explosive ranged healing numbers beyond what even a Medic Archetype can do, and it doesn't even need a roll to get there.
• Slow inflicts a significant debuff to the action economy of a foe that can only be approximated with things such as stunning strike, everyone stepping away from a melee foe, or a Trip that wants the foe to stand back up again - while all of those have their own advantages, Slow can do this for a whole minute instead of once, and - again - from range.
• Air Walk gives something that's usually akin to better flight, granting pseudo-melee-immunity and allowing you to basically just ignore terrain without lingering action costs, earlier than a martial character could - even later, outside of a very select few ancestry feats and costly and limited magic items, it's extremely difficult to gain that inferior flight otherwise.

Wow, those are some incredibly good effects! That could become pretty busted. So what's the catch? On one hand it's limited spell slots, and...
On Enemy Saving Throws
Enemies have good saving throws. Saving throws vary a lot between different stats and enemies, but if you're forcing an enemy of equal level (which is likely stronger than the average!) with a perfectly average saving throw ("Moderate" save in Monster Building Guidelines) to save against your spells, they only have between a 35% and 45% chance to fail it, depending on your level (Odds increase at lv19+). Similarly, martial characters enjoy a 55 to 65% chance to hit with their attacks against the average tough foe ("High" AC in Monster Building Guidelines"). That is a... sizeable difference. How come?
A big part of it is: Spells very rarely actually miss. Most do something even on a successful save, where an attack just completely whiffs - in the white room above, the spell has a 15-5% chance of doing nothing, where the attack has 45-35%. And those save effects are often no jokes either - Fireball still does 3d6 (which is comparable to a longbow fighter hitting), Slow still saps an action. Not big and impressive, but far better than nothing. It really adds up over time.
So much so infact, that the aforementioned fireball - at level 5, where the gap huge at 40% fail chance vs. 75% hit chance - can very nearly match the damage of that longbow fighter in single target damage. The one thing that fighter is meant to be really good at. That's how strong 6d6 basic save is. Fireball is most certainly not meant to be a single target spell. Blasting is good, folks.
On Versatility of Defenses
That whiteroom math just now was probably inaccurate.
It helps to give a rough frame of reference, but the world is not made out of High AC and Moderate Saves - it's made out of all kinds of statistics. Here comes another facet of the discussion: Martials can only use a few skill actions to target things other than AC. Almost all of their main prowess - Strikes - goes into specifically AC. Casters get to target three different saves and also still AC if they feel like it. It can go either way for any given defense, but if you know where to aim for, you get an immediate significant boon to your accuracy on basically any given creature. But how can you do that?
Well, here's a bit of an asterisk: Recall Knowledge or guessing based on descriptions. One is unreliable based on how the table runs it (at least until it gets Remastered in november), one is unreliable based on intuition and curveballs. However, I believe most tables run RK in a way that lets you gather some clues here (and the rest should!) - the high mental stats of spellcasters often also give an advantage in using just that. That action can also be beneficial for martials deciding maneuvers, or can be just as beneficial if they take the action and share their findings. Use it!
(And if you're a charisma caster instead, you're in turn uniquely able to lower those saves yourself via Demoralize and Bon Mot!)
On Versatility of Effects
But that's not all! Not only do different spells target different defenses, but they also do vastly different things. This right here is the true crux of spellcaster power, the rest is just setup. Remember Martials at the beginning: Specialized in hitting things. Really bloody good at hitting things. Solid at some other things if built for it. What can casters do?
Basically anything.
Fireball blows up groups of foes. Slow partially disables a single foe for a longer time and makes their life suck with some teamwork to waste more actions. Pirouette into Heal next round to retcon the enemy attempt to focus fire with no check, Air Walk after the fight to ignore a stupidly slippery cliff or gain blanket melee near-immunity! And that's just four spells. There are so many good effects among them that I couldn't possibly list them all. Yeah, your best tools run out fast and you just don't really have anything that matches the raw power of martial strikes - but depending on your spell list you can make damn near anything else. Walls, forced movement, an incredible array of utility magic that I don't have space to detail more in this post beyond saying it's really helpful, auto-hit magic missiles that outdamage melee fighters against bosses, twenty types of controls and debuffs, a rainbow of damage types for weaknesses, loads of buffs, spells with long durations that you can just cast before facing danger so you get free action buffs, naturally including many, many ways to make said martials much better at using that great raw power they have. Remember kids, if your +1 made the ranger hit, that's your damage!
On Incapacitation
Ah, the dreaded spell trait. Good spells, until a certain enemy level is hit, then they fall off a cliff. A rough solution of wanting "Spells should be awesome and be able to annihilate a foe with horrible conditions" and "Spells should not be able to wreck bosses" in the same game. Many are also single-target, meaning they would be the best in fights against single foes... but that's exactly where they're often useless! Why take something like this over something like Slow, that always works?
Because Incapacitation spells have more powerful effects than those other spells. Doesn't work against the lone boss, no, but might well work against the boss who brought their friends to the party. The conditions these spells inflict are truly horrible and can set foes back so badly. Blindness often more than halves a foe's effectiveness. Impending Doom may be delayed, but comes with -4 AC one round in. Calm Emotions is way up there, taking enemies out the fight like we're playing D&D 5e. Baleful Polymorph actually just a stunlock, Color Spray pretends a Blind and minute-long Dazzle in a cone pretends it's anywhere near putting Frightened 2 on one guy... and remember, if you use top-level slots (as you should), that level limit is not half bad. You can hit APL+0 and on uneven levels even APL+1 foes - you can debilitate foes utterly when they're still just as strong as your entire character combined, or perhaps stronger. In one go. No crits needed. Try them sometime.
On Spell Attacks
So if casters can do anything, what if they pretended to be a Martial? Similarly to how save spells are less accurate, spell attacks are infamous for lacking half damage on miss while still having the reduced accuracy of a spellcaster. They don't even get potency runes! What gives? Short answer, three things:

• These Spell Attacks tend to hit unusually hard, making up for poor accuracy
• Like all spells, they're designed to be situational and only used in paticularly appropiate situations (see above), making up for resource drain
• True Strike

Martials are superior at single target damage, but Casters can spend resources with certain single target spells like most of these to keep up and exceed them briefly with this. Remember - versatility is strength. Copying martials all day is no bueno. But copying martials just a little, for when you could really use another martial specifically right now... that's something that calls for good single target damage, and these might just be what you need. It's not the biggest niche to fill, but it's certainly not one to be completely ignored either!
Here's a graph for one such situation, comparing a few different attacks to a Flurry Longbow Ranger taking (only?) three shots. Target is an equal-level foe with High AC.
https://preview.redd.it/rt4sf5uljv3b1.png?width=678&format=png&auto=webp&s=2bd40648e744b3b7f1ad7bc9fe0c11d24edc7084
The numbers are in direct comparison to the Ranger's damage output - so f.e. the basic Hydraulic Push at level 5 does about -20% average damage compared to the ranger, while the true strike'd version is at +20%. Crits and accuracy are accounted for, so are all the various stat and rune increases (including two elemental damage runes for the ranger at 9 and 15!). However, the ranger arguably doesn't need three actions to pull their shots off, unlike True Striked spells - and all spells here are cast at their maximum available level, so they're more expensive than you might want. Curiously, some of the spell attacks do much better in comparison against those hard-to-hit higher level foes - True Strike scales better than MAP attacks, it turns out. Is this situationalness worth it, when an AoE spell can dish out much higher raw numbers? How many Rangers do you need in your magic beam for it to be worth it? Ultimately, that's up for you to decide. If you don't like the prospects, you have many other great options after the first levels - but they're probably no good match for when you need to hit that one guy over there really bloody hard, right now. (Oh, and a Psychic taking True Strike on an Amp blows past all but one of these curves!)
Wait, you keep comparing to bows. What about melee?
Well, if spells are largely ranged, it makes sense to compare them to other ranged options, right? If you don't agree, things turn out similarly if we replace things with melee martials, except those martials have less actions to spend on attacking, are in more risky positions, and hit significantly harder with the actions they do have.
On Low Levels
All of these things cost spell slots - something that low level casters severely lack. The gap between level 1 and 2 spells is often huge, and where high level casters struggle to earnestly run out of spell slots, low level casters can burn out in just a few combats - early on, even a few turns!
Turns out, cantrips are actually really bloody potent at those. A telekinetic projectile doing 1d6 plus 4 damage isn't impressive, but when the ranged martials don't even get to have damage modifiers, the melee folks have to melee at a level where one lucky crit can be all it takes, and where the martials have no runes yet for superior accuracy... there's suprisingly good utility to that. Especially if you - you guessed it - picked electric arc. Not all cantrips are created equal, and... well, it is what it is, and that is not very well balanced. Healing spells are super good here and Magic Weapon is probably more than a little bit overpowered, and every tradition can access two out of three between these, so casters are definetely able to give impressive contributions even here.
Conclusion
Despite the accuracy and limitations, between all of these explanations, there is one core truth to why spellcasting is good: Having just the right spell, at just the right time, is the single most powerful thing you can possibly do in this game. A fireball doing more raw damage than the fighter will do this entire fight. A Heal coming just in time to save a life, or just a turn of unconciousness, or frying an undead horde. A Slow keeping a boss in perpetual lockdown because they got somewhat unlucky once. An Air Walk that the enemies just... can't respond to. (This is also why the Flexible Spellcaster archetype is actually pretty nifty - you're so much more likely to have just that right spell prepared somewhere.) Keep many tools around and keep your options open. The martials bring home most of the damage to progress those enemy health bars to zero... but when they miss and the enemy crits, when the terrain is screwy, when the dragon has wings and just keeps burning everything down, that's when "I am REALLY good at attacking" shows a distinct lack of plan Bs. Martials do not win fights. Martials progress fights. This is a team game, and if you don't work together, the dragon wins. Casters are the ones that pull the strings in the back and make sure the fight actually progresses in your favor.
And sometimes the boss Nat 1s against Slow. Then yeah, they definetely just win that fight.
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"Okay, I get that spellcasters are useful, but I don't feel useful. They're boring to play!"
Okay, but... how much would you need to feel useful, when you're already being useful? As in, what should the game give you beyond making you actually be useful?
Taste is highly subjective and you're of course not wrong for feeling like that. Nothing is for everyone. But is the problem here within the game, or within perspective? Perhaps the best fix here is shifting the perspective a little - how much did I achieve through my martials or protect them from? How would things have turned out if I wasn't there to make life hard for the foes?
And to everyone else, thank your supports! You've got plenty reason to.
"Okay, but I don't want versatility. I don't want a bag of tricks, I just want to be really good at blowing stuff up and not deal with all of this!"
That I can entirely get behind. PF2e did a great job at making d20-style do-anything magic balanced and mechanically reasonable, but it indeed is built to discourage preparing only fireballs. Perhaps Psychic being more geared towards this kind of thing will help, or Kineticist can fill that niche as a magic-user who's built on a martial chassis, or maybe we'll even get an Elementalist rework in august that makes that useful. Until then, we can only hope.

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