A producer for a band is a lot like a director for a movie. They have most of the say in what the final product is. Sure there is some label (i.e. studio) involvement in a lot of bands (films) but if there is not then the producer (director) has complete control.
The funny thing about this analogy is that in the case of non-pop acts, usually, the writers and the "cast" (band members) are the same people. And in the case of a lot of them, Dream Theater included, they've always been adamant that what they wrote, their vision is what the final product was. With I&W, based on every single demo that I've heard, aside from Oliver's Twist, that pretty much didn't change at all, aside from a couple of riffs that got extended, or little things here and there, but overall, listening to demos for those old songs, even the Charlie Dominici version of Metropolis, it largely stayed in tact, so in spite of all the crap that Prater got, at least he didn't try to change the actual compositions or dumb down what they wrote.
It's with FII that you can hear the REAL difference between the demos and the final product, especially in cases like Hollow Years, Take Away My Pain, Burning My Soul, etc.
But overall, with DT's past producers, I'd say as far as comparing them to "film directors", they've pretty much followed the original "script and the storyboards" very closely. So I guess in their particular case, they'd be pretty useless if they didn't take care of how things sound and the tone of things.
But if we're going to be making film comparisons, then the engineer would be like the Director Of Photography, and they'd be the ones making sure that things look good and taking care of the technical aspects of the camera work, the focus, etc.
So in the case of engineers, they make sure that the final vision is what the producer wants. But I don't see why they can't say, "Listen, this is not going to sound good. Listen to what's happening to the ghost notes," etc.