It's called x86_64 most of the time, but AMD64 is what the original instruction set was known as, so it's kinda kept the name >.<
Just be aware, ia64 (for Intel Itanium processors) is a completely different thing, so if you see that ia64 option, avoid it >.< Other than that, x86_64/AMD64/64bit/Intel-64 all the way
Regarding why: All modern 64bit processors emulate 32bit instructions. This means extra overhead in a ton of different ways. Not only that, but you get more registers (computational memory on the processor) and the ability to address more RAM with 64 bit. You also gain some performance with large calculations. Windows is pretty well optimised for x64 instructions, so you gain there as well.
..Yep.