Welcome Juju, I'm not the one to answer this as I don't know which is better for your children. Warwick is smaller then Leamington and a bit more rural and relaxed, this is either a plus or a minus depending on what you're after. Warwick is closer to the main road to Cov I think (I'm sure someone will correct me if I'm wrong), so commuting would be easier too. Leamington on the other hand has more shops readily available, and more of a social scene going on. But as they're so close together, this isn't really much of an issue.
I hope that at least gives you an idea.
