Q:

What is a Centaur Thief's climbing speed?

The Thief archetype Rogue's Second-Story Work (PHB, p. 97) feature states: 

[...] you gain the ability to climb faster than normal; climbing no longer costs you extra movement. 

However, the Centaur's Equine Build racial trait (GGR, p. 16) says:

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.

Both of these in my opinion are specific, in the "specific beats general" type of rulings, but does one supersede the other?  Would a Centaur Thief climb 40 feet or 8 feet (1+4 extra feet) per round?

A:

From logic, one would arrive at:

"Climbing for a thief no longer costs the thief extra movement,"
"Climbing for a centaur costs 4 extra feet,"
"4 extra feet is extra movement," (implied from "instead of the normal 1 extra foot")
(2&3) "Climbing for a centaur costs extra movement"
C. (1&4) "Climbing for a centaur thief would not cost any extra movement."

A:

The way to reconcile these, IMHO, is to realise that the Thief ability is written assuming that the character is an ordinary humanoid biped. They would pay one extra foot of movement for each foot of climbing, and the ability removes that penalty. 
So you could plausibly claim that a centaur Thief with Second-Story Work pays three extra feet of movement for each foot of climbing, rather than four. That gives them a climbing move of ten feet, better than a normal centaur, but worse than a biped non-Thief. That seems like a plausible outcome within the game world. 
Being a centaur who climbs buildings isn't a very sensible idea, and should not be made plausible by over-literal rules interpretation. 

A:

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.

