Wouldn't really change much. Primary casters like the Wizard would be somewhat worse off, but that's not really a big deal since those are strong classes to begin with. Unless you have a truly enormous point buy, classes like Fighters won't really care since they can get plenty of value out of other attributes.
Will monsters follow these rules as well? The CR system assumes the characters can boost their stats with items to appropriately tackle the challenges they are expected to face.
Just eliminate level up increases to ability scores and use point buy generation (20 points is okay). This way, 20 will be the highest anyone can get without magic. Because of the point buy, anyone with a 20 had to make very significant sacrifices to other ability scores in order to achieve that.
Martials suffer from ability score capping because they rely heavily on stat increases and magical equipment to stay useful.
If you do this. Which I am not criticising. I like this idea a lot.
You should reduce monster AC by 1 across the board and 1 per 4 CR the monster possesses. Even more if you tightly ration magic items. This should compensate largely for the drop in attack bonus your martials will suffer.
Also you might want to invent and introduce a reverse Cloak of resistance for casters to attack saving throws with, possibly in the shape of a staff.
In an effort to produce a campaign where the characters don't seem quite so much like comic book superheroes, I'm considering setting limits on attribute scores. What would be the effects, rules wise, of limiting characters to a maximum of 19 on one score, 17 on a second, and no more than 15 on any other?
According to the CRB, a character with a STR of 18 can lift up to 600 lbs and stagger around with it. By way of comparison, the world's record for clean & jerk, according to Wikipedia, is 263 kg, or about 580 lbs. So I figure that somebody with 19 in any attribute is pretty much at the absolute top of what is humanly possible, and maybe even a bit over.
I would allow these limits to be exceeded with magic, but this will be a low-magic setting, where characters can not readily buy and sell magic items, especially permanent ones.
A clean and jerk is completely different from "lifting a weight and staggering around."
The world record for a deadlift is over 1000 lbs -- depending on the exact method, between 1015-1155 lbs. That would correspond to a strength of ~22, which it's pretty hard to get a human character over without magical means anyway.
Also, I agree -- this hits martials harder than it does casters.