Don't try to get to complicated. There are 2 reasons for having a fine and coarse synchro.
1. Accuracy - exactly the same reason you have hours and minutes hands on your watch. Think of hours as coarse and minutes as fine.
2. Ambiguity resolution - if there should be a power interrupt and the a/c changes altitude by more than 5000ft (one revolution of the fine synchro equates to 5000ft) the coarse synchro takes priority and drives the pointers to the approximate altitude, then the fine synchro drives the pointers to the exact altitude. (this is also the case for very high climb/descent rates)
