Why would such a small, simple program need to be 64 bit? Exactly what changes when moving puTTY from 32 bit to 64 bit? It now takes up more memory?

We are well beyond moving to 64bit on a needs-only basis. At this point it's about dropping legacy and future proofing. There's also no reason 64bit apps should have a larger memory footprint; the ISA and addressing modes are actually much tidier.

Here are the reasons they now provide a 64bit version: http://www.chiark.greenend.org.uk/~sgtatham/putty/wishlist/w...

The most likely reason is ASLR support, since you have more address space available for randomisation. Other than that, you can probably get a slight boost in performance from the differences between x64 and x86, but lose out if the code is particularly pointer-heavy.