It also depends on how you're punched too. Like, typical punch is thumb over fingers, not tucked inside hand against the palm, and typically the contact is made with the flat part of the fingers closest to your knuckles (as well as, obviously, you're knuckles)
But sometimes people can have odd proportions or wear rings or just be capable of punching someone with two fingers positioned oddly that kinda... Makes the punch a staircase surface instead of flat.
When I punch, I tend to unintentionally have my first two fingers kinda... Pushed forward. Which means my punches aren't... Well they kinda suck to be on the receiving end of tbqh.
The bottom image shows how I punch. The first two fingers are pushed out, sort of, and this, actually, is a much... Stronger kind of punch than a flat one like top image.
In martial arts, you typically get trained to make sure the punch you give uses the first two knuckles on your hand because those bones are stronger and the line from knuckle to wrist to arm is straighter which means force is easier to apply and the "recoil" is lesser for you.
(yes there are technical terms for all this, no I do not remember them, I had a shitty night so brain not working 100%)
Anyway. Back to the hitty hitty things.
Punching someone directly, like at the nose, without pulling your arm back to work up a swing, will have a different effect in terms of injury and the force behind it.
If I hit someone across the bridge of the nose from directly in front of them, the force I put in depends on my own speed, accuracy, and existing strength. If I can't pull back my arm to add extra momentum and force, then it's entirely on me and how I land that hit.
A sharp jabbing motion to the bridge of the nose is enough to bruise and possibly break the nose. It will cause stunning pain, the kind that makes you stop for a moment because your brain kinda stalls from WHITE STATIC OW before the aching settles in.
A hit from the side to the nose, as in a wide swing, is more likely to extensively bruise and cause bleeding but less likely to break the nose. Of course it can still happen if there's enough force behind it, because force and momentum science blah-blah.
As said before by @burn-brighter-than-fire when you get a hit to the cheek, it tends to be a throbbing ache. Like, the kind of ache you get after the dentist has jabbed you with needles and bullied your teeth in a root canal or an extraction. It's because the tissue and such gets more blood in it, it causes pressure and swelling which in-turn causes more pressure. You can cut the inside of your cheek on your teeth, bite the soft tissue in your mouth, or even get teeth dislodged or knocked out with a good hit to the cheek.
When it's a punch with those first two knuckles hitting you, as I mentioned with the nose above, there is a higher chance of having your inside cheek getting cut or bit in the act of being punched. The force and pressure exerted with a wide arc swing, the kind you see in fancy slow-motion shots in shows and movies (especially Asian dramas which are đ[okay-emoji]) transfer a lot of energy into a relatively solid area with your jaw and cheekbone offering resistance to the punch you're getting. Bruising is common, a cracked cheekbone isn't ultra rare and if you have a "glass jaw"? Oh boy is that punch gonna hurt even more than it would otherwise!
Beyond the initial burn that has you stagger or stumble with the blow, the kind that makes it feel like you've just slapped a hot cloth to your cheek (genuinely, a punch tends to make an echo sensation of burning, just like a slap, because the skin of your face is very sensitive and the friction alone that's caused by the impact is enough to make your skin hurt), what follows depends on things like where the punch actually landed, where those first two knuckles hit, and the kind of punch delivered.
A wide swing will deliver a lot of force with that momentum. Most people don't know how to take that kind of hit and remain standing because, typically, you don't. Genuinely. Most punches that hit that hard will knock someone on their arse.
I know that personally. From both sides of the fight.
Someone who has training or has learned how to take a hit, can take the force and momentum and work with it. They absorb that force, let it push them back, turn them, shift their stance, even stagger with it and go down to one knee.
Turning your head with the hit will lessen it somewhat but it doesn't diffuse it entirely. Your jaw and cheekbone will take the majority of the force and energy in the punch and it will bruise. It will ache. It will throb. The faster your heart is going, the more it will throb in time with your heartbeat. It will creep up your face, along your temple, and become lime a low-level headache or the echo of a hangover.
Sound will cause that throbbing to spike. Every breath you take will cause the ache to flare. You will open and close your mouth and it will feel like someone is gripping your jaw and making that ache pulse each time you do.
If you're in a fight and have to think and move and talk etc after that punch, it will be somewhat dulled because adrenaline, but you can't ignore it. Especially if you're doing a speech or dialogue etc with your opponent.
If the punch hits higher on the cheek, closer to the ear, btw, you will almost certainly feel intense disorientation. More than the punch would cause because of how our ears work. The inner ear canal, the way nerves of the face are arranged over bone and through muscle and sinew etc. It makes it sound you feel like you're kind of trying to hear sound through headphones that have noise-cancelling effects but also happen to be really tinny and bad with bass and treble. Or like you're listening to something with a pair of headphones that are broke and failing to fully share the audio properly.
That pain, no matter where you get hit, will persist and linger. You will bruise, it will swell, you can easily get a split lip from your teeth because of the punch. So can have that to contend with too.
Basically a cheek punch is always gonna ache and throb with spikes of pain that can be hot and sharp or even make you feel tingly and shocky because nausea and adrenaline are not a great mix.
If the punch to the cheek is more of a direct hit, similar to how you can hit the nose while standing face-to-face and don't have the opportunity or space to pull back and deliver a full swing, the force is obviously going to be lesser and the hit closer to the front of the face rather than the ear and side of the face.
This kind of punch is more likely to cause a split lip and cause you to bite or cut your inner cheek because the force is distributed across your teeth and jaw. Unlike a hit to the cheek that comes from the side, there's no "hollow" space for the punch to strike so all the force in the punch gets despised through solid bone and teeth.
Honestly, I'd prefer a punch to the side of the face that makes me feel like I'm underwater with my hearing and a mega hangover over one that mainly hits teeth and my jaw. Genuinely. You can more readily take that kind of hit and stagger with it, adjust your stance to keep your balance etc.
Any punch to the front of the face, or near front, tends to just make you go backwards because your stance and balance tends to not be able to adapt to without giving your opponent more openings to then follow with more hits.
A step backwards as you stagger makes you more vulnerable (in my personal opinion and experience usually) than a step sideways or a stagger to the side because all your weight shifts back and that shifting can be harder to correct or compensate for in a fight, especially if you're disoriented or already injured.
A nice little aside here, specifically for a cheek punch is that if you have your jaw relaxed and muscles slack when you get punched, you will not be as injured. Tension is what makes a cheek punch worse. Tense jaw? Energy transference will make it bruise worse. Tense muscles of the cheek? More likely to cut your inner cheek and get worse bruising, maybe even a fractured cheekbone.
Same as with the cheek punch, if it's force that comes from a side punch, as in a wide arc swing, the force and momentum is gonna really turn your head. It's basically just a blunt force trauma to your temple. Disorientation, balance issues, heck, consciousness issues are expected here.
The pain is sharp and striking, can be easily like static and if I had to assign a colour to a temple injury/punch, then it'd be silver-white that just disrupts everything.
A hangover, a headache that presses and presses and presses is the easiest kind of explanation of the pain a temple strike causes. If you've ever grabbed your head and pressed the flat of your palms to your temples and felt that? That's sort of how it feels. It's a vice that just causes pressure that turns to an ache to a tingling, electric pain. Because blood builds where that punch landed and pressure builds under tissue and skin and it doesn't let up.
Add in a pounding heart and adrenaline and you get a baby migraine that makes following conversation agony. You get lights that stab your ears. Sounds that gouge your eardrums. You get everything aching even if you're not injured anywhere else because your brain is getting pressured and that has to translate to pain, it just so happens the pain ends up feeling like it's everywhere.
And with a temple strike, it doesn't really matter what kind of punch you get. The results tend to be the same. An upper cut, a full swing, a direct single punch with no pull-back and swing? Doesn't matter.
Any punch to the face hurts, for sure, but I'd rather not get punched in the eye tbqh. If the punch lands with those first two knuckles, you can easily get fractured eyesockets or your nose broken or a cheek fracture. If it's not the first two knuckles it still can do all that.
A pull-back and swing punch will bruise, make a stinging burning, an audible crunching or cracking if bones are broken or fractured, and that eyeball is probably going to be useless to see out of for a bit because blood and pressure and pain.
The pain can be sharp for a moment and subside to a throbbing that goes with your heartrate or it can be overwhelming because things break under the force exerted.
Unlike the cheek with the bit of padding and "hollow" space, the eye is in the eye socket which isn't a super large space, is hard and not padded like the cheek, jaw, and mouth, so when that full punch lands?
Bones can bruise just like muscle and tissue.
Bones can also fracture and break.
A fractured eye socket is... Its agony in the moment and blinding for a while after. Then it aches and throbs and stings periodically while it heals.
You know how your brain can sort of... White out for a moment when you elbow something? Because, you know, that's your "funny bone" and the nerves get all jangled for a moment? It's because your brain doesn't know how to process the pain for a moment. Then it hits.
If it's a little bang or bump, it stings like a shock and then fades to a throb like a distant rumbling of thunder barely audible to you.
If it's a bigger bang or bump, a jab, or even a fall with all your weight on the elbow joint, it's still a shocking sting but it turns into electricity in your nerves, up your arm both directions, aches and throbs and feels like the impact point is being pinched and twisted over and over until it eventually fades.
Getting punched in the eye is sort of like that.
Your nerves get overwhelmed because they're close to those bones and intertwined with muscle and sinew and ligaments and tendons and so on. That crackling electricity whites out everything for a moment, a split-second perhaps, before it comes back as your brain processes and translates and turns that input into pain pain pain.
I have been punched and kicked and slapped in the face many times before. I've blocked strikes that would have easily knocked me on my arse and caught one's that did. Of them all, getting hit in the eye is honestly the worst.
Because you don't just have the pain to deal with. If you're unlucky, you also have the panic and terror that can follow when you think you're blind in the eye. Because your brain stalls for a moment. It literally stalls from that pain input it has to process and translate and then send back out.
I have had the displeasure of being punched in the eye and felt for a long moment after, easily a few seconds (which can truly make a difference in a fight) that I was completely blind in the eye that got the hit. Because again, as mentioned, the pressure and force sort of... Makes a suction issue. It's like a mini implosion as the socket is covered and your eye pressed back and also pushed forward simultaneously.
I don't know the fancy science for it but I found that it seemed to have... Almost caused my eye to Error 404 for a hot minute after the punch for whatever reason. I literally couldn't perceive anything with that eye for about ten seconds (probably less actually but to me it felt like a lifetime for a split-second of disorientation turned confusion turned fear) before it subsided and I could see again.
My vision was blurry and took a good couple blinks to return to "normal" and even with the pain becoming less every second that passed, I genuinely would prefer to be punched anywhere else on the face than my eyes.
This isn't universal to everyone and their experiences, obviously, and I'm coming at this from the side of how the punches can affect the pain and injuries but
TL;DR: if you ever have a choice of where to get punched, go for the side of your face for the cheek and keep those muscles slack. A punch to your eye is the least preferred imho 0/10 do not recommend.