What is the difference between 32 bit and 64 bit launchers of Transistor?

A 64bit program can not just address more memory than a 32bit program, it can also use some CPU instructions which are only available in 64 bit mode. That means a 64bit program can perform certain calculations in one step which a 32bit program has to split up into multiple steps, which takes some additional time.

This, however, only applies when:

1. the compiler (the program used to translate the programs sourcecode into an executable) used by the developer knows how to make good use of 64bit instructions
2. the most CPU-intense parts of the program even benefit from the new 64bit instructions
3. the program is mostly limited by the CPU performance and not, like most games on most PCs, by the performance of the graphics chip.

The 64bit architecture also has a drawback: Some pieces of information which are stored in 4 bytes in 32bit mode now take 8 bytes. This means a 64bit program will take a bit more space in RAM and (more importantly) use more CPU cache. That additional information also needs to be moved between RAM and CPU, which puts some additional strain on the memory bus.

Still, as a rule of thumb, when you have a 64bit operating system, and you get to choose between a 64bit and a 32bit version of the same program, you should generally use the 64bit version.