"Build out" versus "build 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.

As part of the build-out plan, the landlord was expected to partition the west wing into 14 separate offices.

To "build up" means to make something more than what it was. It may be used in a literal sense

You can build up your muscles by exercising with free weights.

or it may be used in a figurative sense

Praising children helps them build up their self-esteem.

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.