It all depends on what kind of shot you get on a deer, if you're hunting deer with a 22 LR, then you would have to go for an upper neck shot, thus severing the spinal chord, resulting in a quick kill.

However, if you do as most hunter's shoot, a vitals shot, most of the time, not always, but most of the time, after a vitals shot, the deer will run, and you will have to track it.
While the deer is running away, I bet you he would feel a difference, a deer hit with a different bullet will bleed differently, the expansion of the bullet will vary from bullet to bullet, that's inevitable, so, yes, a deer, would feel a difference between different bullets.

Of course, in real life, you would be right, a deer wouldn't feel the difference, simply because, it wouldn't survive to feel one and then compare it to another, most deer never live long enough to be shot on two seperate occasions, as most wounded deer die at least a week or two later from infection.
There are exceptions, though, but this is more often an occurence with bowhunted deer.

A bigger, more powerful bullet, if shot at a deer, from the same distance, same spot, etc., would prove to be more effective than the smaller bullet.

Let's just use the example of the 30-06 and the .270, both have the same powder capacity, but the bullet size is different.

Of course, in the field, it's all about the shooter's ability to shoot well under thepressure of buck fever.
I'm just talking as if it's a lab test, but the broad debate that was raised, was whether or not a bigger bullet would have better performance on a deer, versus a smaller bullet. I say yes, and no offense, but the fact that your grandad could kill deer with a 22 has nothing to do with whether or not bigger bullets can be more effective or not,what you were saying about killing deer with a 22 is another debate entirely.