First, you shouldn't pick one over the other. Let Windows make that decision when it installs the software.
Like most up-to-date PC users, you're clearly running a 64-bit version of Windows. Computers capable of running 64-bit code (also referred to as x64) have been around for almost a decade, and are pretty much ubiquitous these days. The same goes for x64 versions of Windows.
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But although the vast majority of PCs these days run x64 Windows, a great many programs (quite possibly most of the ones on your PC) are still written for the older 32-bit version. For purposes of backward compatibility, Windows x64 needs to run both 64- and 32-bit programs.
Windows runs smoother if it keeps these two very different types of code separate. The operating system can't assume that an x86 program even knows that such a thing as x64 code exists, and that could cause problems if they cross. For instance, if a 32-bit program went looking for a .dll, and found one that came with a x64 version, the program wouldn't work and wouldn't know why it didn't work. Keeping them in separate folders is the simplest way to avoid such problems.
So why is 32-bit code identified as x86 instead of x32 The 16-bit chips in early PCs used the 8086 architecture. Even when the chips went 32-bit in the late 1980s, they still used 8086 code, and x86 model numbers. (Remember the 386 and 486 processors) So the number 86 now refers to pre-x64 code, whether it's 16- or 32-bit, although the 16-bit x86 code won't run in 64-bit versions of Windows.