Mounting Sub in Box

to answer the actual question here: "so... does it make a difference in performance? If so, is it a big difference?"

The answer is yes. it will change the sound. Much? Probably not. The difference is only in that 1) the volume of the box will change when the extra displacement of the woofer magnet and basket is removed. Depending on the sub, it could be as much as a 0.75ft^3 difference. Depending on the box size and style, this could be a fairly big difference. I would assume your sub has a displacement of about  0.5-0.75 cu ft.. If you are using a sealed box of about 1.0-1.5 cu ft, a change of .5 cu ft. could make a big difference. Expect the sound with the basket out (bigger box) to be deeper, boomier, and looser. Expect it in the smaller box to be a bit tighter, punchier, faster, and not quite as deep. If you are using a ported box, the change could have a bit more of an effect, because the tuning of the box will change more dramatically. Will it be a big listening difference? Probably not.

2) You will be hearing the bass more form the back side of teh cone with teh woofer out. Dows this matter? In a sub, not really. A very detailed listening might find some difference, but in a trunk, there will be no real  way to tell from the cabin of the car.

Best bet: Try both ways. If you can tell a difference, use the way that sounds best. If not, use the one you think looks best. That simple.