For a game, the principal difference is access to RAM. Win32 programs are limited to 2GB of memory, unless they use tricks like the 3GB compilation flag, PAE and such that allow more memory access at the cost of performance. Win64 compiled versions have access to a memory space far larger than anybody is going to have in a home PC. There's some other stuff as well, but memory is the most important part. 2GB isn't limiting when your other platform is the 512MB equipped PS3, but much more limiting when you want to do 4k graphics and your other platform is the 8GB PS4.
Even running a Win32 compiled program (like the same version of the game) on Win64 gives you 4GB to work with, because due to how the virtual memory system works it can allocate the entire 4GB address space to the game instead of needing some of it for the OS and hardware. With the PS3 gone, Win32 PCs are now their least common denominator platform. People started shifting to 64 years ago and it's fairly common now, to the point that among Steam users, Win32 is ~2% usage. Could be higher amongst FFXIV users, but you pretty much can't buy a Win32 PC anymore unless you're an enterprise level customer.
The saddlebags are interesting because they've cited data transfer of character information between the world server and the instance servers as a reason for not having larger inventory multiple times. The saddlebag inventory is specifically not available when you're in an instance, which suggests to me they're adding that instead of more inventory because it lets them avoid the extra memory required on the instance server (in a dungeon you simply don't have access to that inventory so that server doesn't need to know what's in it).Yeah, I'd love for them to do it all in one go, but I don't see that happening. They'll update the way they did with SB and probably some unnamed patches. My guess is that's where the saddlebags are coming from is a small update on a part of the old infrastructure.
The glamour closet is the same thing in that it's also only available in specific locations and thus doesn't load the other servers at all.
(I am a programmer, although I don't know what their backend looks like so some of this is speculation.)