What's the difference between an Engineer and a Producer?
The engineer is the "equipment operator".  His expertise is choosing the right equipment for the job and using it in the right way to achieve the desired effect.  His role is analogous to that of the cameraman in film production.

The producer is kind of like a project manager.  His job is to get all the individuals - musicians, vocalists, engineers - to each do their part to achieve the overall goal.
For example, he might decide a certain song should be quiet and soothing, so he'll advise the musicians to tone down the accompaniment, the vocalist to soften her voice, and the engineer to add the necessary processing to achieve that soothing sound he's after.