The special edition DVD features the alternate ending where there is a different dialogue after they both kiss in the car, during the final scene. Richard Gere then steps out of the car (very emotional scene) before walking up and entering the police station. Fox & Regency wanted this ending at first (it was more Hollywood-esque, in-your-face, clearer kind of ending), but the director and the cast insisted that they put in the current ending you see on the theatrical release.
The scene where Connie rides the train home after her first sexual encounter with Paul, and silently remembers the encounter, was filmed in one continuous take. The camera simply stayed on Diane Lane as she went through a series of expressions, and then the scene was cut and edited together.