x86 or x64 - what is faster?

Depends who's writing the program.

But in principle, and provided everything fits within the 4 GB address space limit, and the same hardware can handle either mode, x86 should be quicker because the memory transfers will be smaller.

It depends. Some projects only have 32bit executables though as 64bit CPUs are more recent they are likely to be faster. I don't know if anyone has tried using a 32bit and 64bit version of BOINC and or OS to compare speed. I would guess there is an advantage to 64bit if the executables are compiled for it.

x86 or x64 - what is faster?

At what?

It depends what you’re trying to do

64bit can handle more complex instruction sets.
Your question is like asking, what is faster, a basic calculator that can do only sum and subtraction, or a scientific one that can also multiply and divide?

x86 would be faster at what it can do (like the basic calculator that can only do sums and subtraction).
x64 (comparing here with the scientific calculator) may be slightly slower at sums and subtractions, but is massively faster at multiply/divide and other functions.

So for a project using only sums or subtraction, the older calculator may be the better thing.
Likewise, unless a project uses x64 instruction sets, x86 instruction sets will be quicker.

x86 or x64 - what is faster?

In perfectly optimized code x64 usually be faster due to more registers available.
(Modern best optimizing compilers good enough to get measurable speedup for x64 build over x86 one on computationally heavy apps).

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Also the better addressing of large memory spaces can help when pulling data from memory rather than disk

Also the better addressing of large memory spaces can help when pulling data from memory rather than disk

Thus encouraging bloatware, instead of mean and clean programming - and slowing things down again.