The real issue is parallax. Most centerfire scopes have parallax set for either 100 or 150 yards. A rimfire is typically used at a much shorter distance. It either needs adjustable objective or side focus to bring zero parallax point down to your shooting range, probably 50 yards, or a rimfire specific scope that has fixed parallax set at 50 or 60.
When parallax is off, unless you get your eye exactly, I mean exactly, centered in the eyepiece and do that on every shot, where your crosshairs are aimed may not be where the bullet goes. Groups open up or small targets (squirrels, etc.) get missed for no apparent reason.
Many centerfire scopes can be adjusted for parallax, although many people do not know. For instance the Leupold VX-1 has a screw on cover on the front objective. If you unscrew that you will see that there is a second set of threads that permit you to slowly rotate the objective lens in or out, as you would if it had an adjustable objective. You can look through the scope and slowly move that objective lens, a quarter turn at a time at a target at the distance you want your parallax set until it is both in focus and you can move your head a little bit off center of the crosshairs and they stay on the target. Then screw the outside cover of the objective back on. That is the only difference between the rimfire VX-1 and VX-2 and their standard centerfire counterparts, but it is an important one.
Unscrewing the outer cover does not break the internal seal. Under that cover is the second threaded housing that holds the objective lens. This one has slots. You slowly rotate this in or out to adjust parallax distance. Do not unscrew it all the way or you will break the internal seal, the nitrogen, argon or other inert gas to escape, and allow moisture and air inside the scope. It would have to go back to the factory to be purged.