A premise is an idea for a movie. For instance, a killer shark holds an island town hostage over the Fourth of July.
However, a logline is a concise statement that summarizes what a script or film is about. In keeping with the above example, it would be: A police chief with a phobia of open water battles a killer shark with an appetite for swimmers and boat captains, in spite of a greedy town council who demands that the beach stay open.