A knife actually delivers more energy to target than a small caliber bullet does...
A typical knife attack is 7-10 m/s
†, a typical knife 1 kg... but the knife also has the mass of the arm, unlike a bullet, and a typical arm is 4-6 kg... using the minimums and maximums, 98-300 J. Putting one's full mass behind it at a run can generate up to 20 m/s and 80+ kg... 16 kJ... (E= 0.5*M*V²)
A .22 LR (≅5.5 mm) is 155 to 190 J
at the breech, and a .45ACP 350-495 J
at the breech. ‡ actual energy on target can be half or less, depending upon range.
5.56N runs 1790 to 1890 J, and 7.62N 2990 to 3590, again, at the breech. ‡ That's the energy in the powder on a standard specification round. The barrel steals some air steals a good bit more, some is wasted on too-short a barrel... if you have muzzle flash, the round isn't getting all of it.
300 J
on target is comparable to many small-caliber pistols, and a good knife has less contact area (0.1 x 30mm is 3 mm² for a good knife's
initial tip penetration; πr² on a 5.5mm is around 23.75 mm² for the light pistol... penetration is better for knives; knives do actually shed almost all their energy in target, and friction increases rapidly with depth, and small caliber bullets do tend to go through on extremities, which also means not all the energy is in target.
At the level of detail in the QS, same damage for both seems pretty reasonable, because the energy dump into target is in the same range.
Now, armor seems pretty useless RAW.
†
https://arxiv.org/ftp/arxiv/papers/0811/0811.3955.pdf
‡
3g³ Guns! Guns! Guns! by Greg Porter.