There are several benefits from going 64-bit. Firstly a 64-bit operating system can use more than 4GB of memory in total, while a 32-bit OS is limited to 4GB (and as FalconFly noted, for reasons that are too technical to describe here often only 3.5 or 3.25GB of that is actually available for use). The second main benefit is the memory available to each program/game you run; 32-bit programs are limited to 2GB of memory. This is the case on both 32-bit and 64-bit OSs, so the current build of Elite Dangerous can't use more than 2GB no matter how much memory you have fitted or what version of Windows you have.
A 64-bit program can in theory access up to 16TB of memory, and in practice will let you use 128GB+ unless you are running the Home Premium version of Windows 7 which is limited to, I think, 16GB of memory in total.
This is the reason why games are going 64-bit so quickly now. The 2GB limitation on 32-bit is becoming a major pain, games have to keep loading in data from the hard drive while you're playing because there isn't enough memory to load it in advance. Stuttering, pauses, long loading times, and lacklustre texture detail in games are often down to the developers having to squeeze everything into that 2GB. And some features that developers want to do just can't fit in 2GB. Multiplatform games have particular issues here because the new consoles generally have 5-6GB of memory available for use, so keeping the PC build below 2GB can cause huge problems.
tl;dr - 64-bit is good and makes for better games. I'd expect the majority of high-profile games to be 64-bit only by the end of next year, so anyone still on 32-bit Windows is going to have lots of reasons to upgrade.
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