Specific Beats General Still Applies
Second-Story Work’s description is:
[...] you gain the ability to climb faster than normal; climbing no longer costs you extra movement.
This rule applies to any climb you make, whether that is up or down a vertical surface or object, such as a rock wall or a rope, or if you are climbing along a horizontal surface, such as a cliff edge or monkey bars.
The Centaur’s Equine Build’s description is:
In addition, any 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.
However, if a climb requires both your hands and your feet, this second rule comes into play. This is because a “climb that requires both your hands and feet” is a more specific term than just a “climb”, so specific beats general.
A vertical climb, more often than than not, requires both your hands and your feet to climb (your hands to reach up and your legs to push you). However, a horizontal climb may only require use of your hands and not the use of your feet so this second rule would not come into force. Using the example of monkey bars, your legs typically dangle downwards and you swing your body forwards to grab the next bar, they do not require use of your feet, only the use of your hands to grab the bars and your body to swing you forwards.
Additionally, any climb that only required the use of your feet, such as climbing stairs, or hills, or rocks or any other uneven surface. As long as your hands weren’t needed for the climb, the second rule would not apply.
So then, if a climb required both the use of your hands and feet, you could only climb 8 ft (1ft + 4 extra cost). If however a climb required only the use of your hands or only the use of your feet, then you could move 40 feet.