I got Fire Emblem Awakening and I'm considering playing on Casual mode because I don't want to restart a chapter each time a character dies, but I don't want to miss any unlockable either. Are there characters or gaiden maps which are only on Classic mode?
No, you will only miss out on one thing by playing on casual mode. If you beat the game while playing casual on the Lunatic difficulty, the NG+ for Lunatic will be casual only. This is different from beating it on classic mode which unlocks both. This is also different from the other two difficulties where beating it in casual unlocks both classic and casual NG+.
Other than that the games are identical except in casual mode permanents deaths are turned off and you can save in battle, you're allotted two slots for this. There are no extra boards/characters/etc. that can only be obtained by playing on classic.