I think it's given away in the name - in a backline setup, the inside center stands closer to the flyhalf than the outside center. Some inside centers (eg Billy Twelvetrees) play almost as a second 10. They pretty much take on the responsibilities of a 10 but further down the line. They may take on kicking duties eg when the flyhalf doesn't have enough time to get a kick away. But mostly they need to be good distributors. Don't see these guys much at 13.
Ball-crashers (eg Tuilagi, Roberts) can play in both positions. Since defenses are more organised the closer to a ruck you get, and help in recycling comes faster as well, it seems to me as if ball-crashers who play 12 are mainly used to create space on the next phase by taking the ball with speed, running straight, hitting the gainline, and tying up defenders. Since at 13, the gaps between the defenders get wider, tacklers start to find themselves more likely to be one-on-one, and a good ball-crasher at 13 will often use their strength to flatten players, hand-off players, draw players and then pass it down where more space has opened up etc., to create an attack on the current phase. Because ball-crashers at 12/13 primarily use their strength, they seem to be somewhat exchangeable.
Similarly, faster/more intelligent players can achieve the same results as a ball-crasher at 13. They'll often run at an angle around the outside of a player, especially if that player is trying to hit the inside shoulder (the shoulder of the attacker closest to the ruck), and if they do it quick enough, they can get around that player. It's about tying/drawing defenders to them so that their wingers have more space, and I figure them to be the back that will punish defenses the most. These guys don't seem to play 12 much, but often have secondary positions in the back 3.