64 bit is become bussword, and mean many unrelated things.
Basically it was about CPU and MMU address register size that can be directly operated, w/o any tricks, and mean there is hardware unit in CPU that can handle 64 bit oiperations with ADDRESS. in short, it allow you to operate very big arrays of data w/o any segment tricks, like nightmare in 8/16 bit era.
But recent CPU usually do not only have 64 bit ADDRESS operations, but increase general register size, incerase register number, and add more instructions to better handle that. It is orthogonal with address, dont mix them. Technically, there is possible to make CPU that can do 64 bit general operations (like C = A+B ) but have 32 bit addressing or lower. And oopposite, can address 64 bit data but have 16 bit ot 8 bit general registers. Actually, mainstream Intel and AMD CPU can be configured to any combination, for compatibility reason for example.
So, if you going to jump to 64 bit OS like Windows 64 or Linux 64, you get both 64 bit address, 64 bit general register pool, and new modern instructions, all in one. You get same time less restricted memory access, more and wider registers (good for complex programs, less memory access) and faster mor modern instructions. For example, x86_64 platform (recent AMD CPUs ) assume SSE2 instruction set for float point operations, it much faster then old archaic FPU 87 coprocessor instructions, and differ very lot.
There is possibility in very corner cases to get slower progrems, if it very depend on pointer size, as now pointer need 2 times more memory, 8 bytes instead of 4, but in reality such programs was rewriten long time ago and i ill be very surprised if you get any real program that will run slower on 64 bit OS.