There's very little difference. As some others have pointed out, there are some differences in pronunciation--in classical Latin, c and g are always hard (like k and the g in "get," respectively); in ecclesiastical Latin, when they occur before e and i, they're sound like the English ch and j, In classical Latin, v was pronounced as w; in ecclesiastical Latin, it's pronounced as it is in French, Italian, and English. Yes, pronounced like modern Italian.
The grammar and syntax of ecclesiastical Latin are somewhat simpler than those of classical Latin. Someone with a good grounding in classical Latin will probably notice simply that ecclesiastical Latin is easier to follow and translate, without even noticing what the differences are.
In short, if you're learning classical Latin, you'll be able to read ecclesiastical Latin with NO trouble! I'm surprised that anyone who has taught Latin would say otherwise.