How are these two different?
Build [something] out
Build [something] up
in U.S. English at least, build out usually refers to finishing the interior of a building or an apartment that one is renting or leasing.
To "build up" means to make something more than what it was. It may be used in a literal sense
or it may be used in a figurative sense
As an American, I can't say I've really ever used "build something out" much, though perhaps a Britain would use it more frequently and could tell you better. If I had to guess as to its meaning, I would assume that one means to expand something perhaps?
To build something up is to make something sound fantastic. That I have used a lot more frequently than to build something out.
Build out: Building horizontally
Build up: Building vertically
That is how I would interpret it, and would use it, however rarely is "build out" used, and when it is, almost explicitly in the physical act of building something.
Building Up can be used figuratively for improving anything.