And then there are "breakdowns" which if I remember correctly are somewhere in between layouts and full pencils.
Breakdowns and layouts are essentially the same thing, or are so close as to not really make much of a difference, though some artists can be very particular about what constitutes what. At that point, though, it's pretty much just splitting hairs, if there even are any hairs to split.
To answer the original question, though: "finishes" is simply a term used when you're also using the term/s breakdowns or layouts. You really wouldn't see credits that include a penciller and a finisher at the same time.
To understand the difference between layouts and pencils, remember that pre-ink comic book art really occurs in two main phases: the layouts, where the artist chooses the camera angles, positions the figures (who at this point are just blank bodies), anticipates where the word balloons might go, etc.
The finishing phase is where the artist goes back and fills in the specifics--turning ovals into faces, geometric blocks into computer consoles, detailing the uniforms, etc.
Keep in mind that these are distinctly different skills. A good breakdown artist has to really know sequential visual storytelling, how to create pacing and tension in the shots, how to move the reader's eye around the page, etc. The finishing artist is where what most people think of as the "style" comes in--in other words, why Frank Miller's art looks like Frank Miller's or Jim Lee's looks like Jim Lee's.
Some artists have a great visual style, but are desperately lacking in sequential storytelling skills--and that's when you see comics that look like a series of poses and page pinups when they handle all the pencils themselves.
