Actually, motorwise not much. the main difference is in the layout of the bike. the XL1200C (the Custom) has a different rake by a few degrees. Its more of a sport cruiser. where the Sportster Sport is a sport bike. With a smaller gas tank, a more upright riding posture, mid position for the footpegs. And the Custom has a bigger gas tank, forward control position on the footpegs, and a more laid back riding posture.
Dont let people tell you that a Sportster is a "girls' bike. My last Harley was a 1957 Panhead, and it had the same size engine in it that my Sporty does now. a 1200cc or 74 cubic inch engine. The Sportster is still quicker than most of the "Big twins", and after 2004 the engines were rubber mounted. Let me tell you, that made a lot of difference. Its nice to look in the mirrors and actualy see what is behind you.
Hope that helps.
I own an '05 XL1200R the modern equivalent to a Sport, and my brother owns an '06 XL1200C XL1200C