Swapping out the GPU only really applies to desktop computers (yes, there are exceptions, but small ones). AMD and NVIDIA are separate companies, you can't CrossFire an AMD GPU with an NVIDIA one, nor can you use SLI with an NVIDIA GPU and and AMD one.
Differences
well, you can either get a really technical explanation or a really simple one. Simple one: different chip designs, different design execution, some software performs better with an AMD GPU, some with NVIDIA. Depends on how they're optimized.