Chamois in your shorts help prevent chaffing(sp?) during long rides.
Yes it does make a difference but i wouldn't bother for 1 mile.
Infact i wouldn't bother with any bicycling clothing for 1 mile other than a helmet.

Also, you'd probably get better excercise if you just walk the mile. It would be a lot cheaper and would keep your heart rate up for longer if exercise is your goal.