Shot placement above all, followed closely by a round that will penetrate deep enough to go through the CNS or major heart/arteries/lungs (in order of best-worst.) followed by lots and lots of holes.
In my experience a pistol round, even a "Big" one...is a really loud icepick. If you stabbed something with icepick A versus icepick B which is 2.5mm wider, would it overly care? Probably not.
Would it care if you stabbed it 15 times versus 7? Yes. A lot.
Would it care if one stab was through the aorta versus one to the lung? Yep.
Rifles? That's another matter. You start seeing enough velocity to create a temporary cavity large enough to rip flesh and structures by overstretching, along with bullet fragmentation leaving you with *slightly* more room for error. Is it more lethal a hit with a pistol to the same not-immediate-
KO spot? I can't say, but the survivors tend to be a more messed up long-term.
its the relatively killing potential between 7.62 and high velocity 5.56 that I don't know about.
Speaking of US military ammunition only:
7.62 does not fragment reliably at any range. 5.56 does at close range.
M855 5.56 can shoot through personal armor that will happily stop M80 7.62 (Particularly things made of steel close enough that the 5.56 still has a velocity advantage.)
This is also before you factor in better trajectory, shorter flight time, and lighter recoil into the mix...to say nothing of the platforms firing either cartridge. All else being equal? Gimme the 5.56, frankly.