Using a climbing speed unambiguously overrides all rules related to climbing without a climbing speed, and therefore all game features that adjust those rules; eliminating the penalty for climbing without a climbing speed does not replace said rules, but overlaps with them in an ambiguous manner
The new Centaur PC race's Equine Build trait create an ambiguity for cases where "Climbing doesn't cost you extra movement" that does not exist when you have an actual climbing speed. The relevant part of Equine Build is:
[A]ny climb that requires hands and feet is especially difficult for you because of your equine legs. When you make such a climb, each foot of movement costs you 4 extra feet instead of the normal 1 extra foot.
If you have a climbing speed (e.g. granted magically via Spider Climb or the like), then there is no ambiguity. The incremental penalty applies to using your regular speed to climb, but if you're using any climbing speed you've been granted, you're bypassing the rules for climbing without a climbing speed entirely; you'd just use the climbing speed directly. "Specific Beats General" applies here; there's a general rule for climbing without a climbing speed, which is modified to make it worse for Centaurs in that situation, but that general rule only applies if you lack a climbing speed, so the rule (and all modifiers to it) don't apply.
But if you only have a feature like "Climbing doesn't cost you extra movement", you don't have a climbing speed, and you're still using your regular speed to climb, so "Specific Beats General" is not obvious doesn't apply. The general rule is:
- Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you're climbing
and you have two specific features that modify that general case in opposite ways:
- Climbing doesn't cost you extra movement
- When you make such a climb, each foot of movement costs you 4 extra feet instead of the normal 1 extra foot.
There is no hierarchy of specificity nor any order of operations for applying these rules, so there's no unambiguous interpretation. In practice, I suspect the writers of Equine Build made a mistake, and instead of providing a general rule, like the (kinda ugly) "Climbing using your normal speed costs three additional feet of movement per foot traveled, which cannot be eliminated by feats and class features which eliminate the extra movement cost of climbing", they described the common case of climbing non-difficult terrain without such features, ignoring the edge cases. By describing an example case instead of the general rule, there's no unambiguous way to handle a centaur:
- Climbing a surface that counts as difficult terrain. The example in Equine Build only talks about increasing a 1' penalty to 4' penalty, what if the non-Centaur penalty is 2'? If we have to rely on the general difficult terrain rules, not the specific rules for climbing that clarify the penalty is additive, not multiplicative, maybe you're supposed to pay a 4' penalty, then halve your speed?
- Climbing without a climb speed but with a feat/class feature that eliminates the climbing penalty. Do we apply the racial feature first, then the feat/class feature, arriving at no penalty? Do we apply the feat/class feature that eliminates the penalty first, then apply the racial feature? If so, does the racial feature restore the penalty to the full 4' or just add 3'? With no hierarchy of specificity, it's a DM judgment call.
That's all I can give you, rules as written, but my preferred ruling (below) would lead to a situation where eliminating the penalty for climbing with your normal speed behaves differently from being granted a climbing speed equal to your speed.
How I'd rule:
My personal approach here, as DM, would be to do my best to incorporate both the bonus and the penalty as appropriate. If your centaur has Athlete or Second Story Work (but not a climbing speed), I'd interpret that as eliminating the normal penalty, but not the extra penalty from Equine Build, so you'd pay a 3' penalty for a climb requiring hands and feet (4' in difficult terrain). This ends up with a fairly reasonable end result:
- Humanoid with 30' speed and no special climbing features: Climbs at 15'/round
- Humanoid with 30' speed and base climbing penalty eliminated: Climbs at 30'/round
- Humanoid with 30' speed and a climb speed equal to their speed: Climbs at 30'/round
- Centaur with 40' speed and no special climbing features: Climbs at 8'/round
- Centaur with 40' speed and base climbing penalty eliminated: Climbs at 10'/round
- Centaur with 40' speed and a climb speed equal to their speed: Climbs at 40'/round
Case 2 and 3 (the common case) don't differ meaningfully, because eliminating the penalty gets the same effect as being granted a climb speed equal to your speed. But cases 5 and 6 are very different; special non-magical climbing training is not as good as being Spider-centaur.
Assuming that true climb speeds (that aren't a racial feature) are only granted by magic (which would make sense if they were careful to ensure non-magical climbing benefits are not granted as an actual climbing speed, like they do with Athlete and Second Story Work), this makes sense. With no magical assistance, trying to climb with four inflexible single-toed hooves instead of flexible multi-toed feet is nigh impossible (they're being pretty generous not requiring you to be winched up every sheer vertical climb with a rope and pulley). Special non-magical training makes you a little better, but there are limits to how much you can train yourself around not having proper footholds. But if you're magically granted hooves that stick to any surface (so even a humanoid wouldn't need to grip with their toes, they can just walk up the wall), you can do the same, and can (if the climbing speed is based on your normal speed) do it even faster, cantering up a vertical wall just fine.