Noir is a pretty general genre, in fact it's more of a stylistic gesture over the film's entirity. Since Noir is so applicable to so many genres, it's not so nailed to one thing, even though it's predominantly used in Crime Thrillers, and such. The Noir ideals are to use contrast colours often, not in terms of bright vibrant colours clashing, but use of shadows, and light in the sense of blackness, and whiteness, and usually take a cast of characters, who similate qualities of "Anti-Heros", who all have their justifiable reasons to do what they do. Noir, essentially, takes grim colours that contrast, dark story line's, characters with villainous qualities, or Anti-Hero qualities, and centre around the content at a slower pace than the usual film.
Neo-Noir is a very similar concept, but was more of a revitalization of the qualities that the Noir Era films gave the world, but updated them to make up for the changes in film, such as the introduction of colour, even though some late Noir films were in colour. Essentially, Neo-Noir is a style that simply applies the basic principles of Noir, and tweaks them to update it in a sort of way. A great example of a film, which carries a lot of Neo-Noir style in it, and goes unnoticed by a lot of film lovers, is Pulp Fiction, since it carries a dark interwoven story, a cast of Anti-Heros of all sorts, and a large amount of contrasting light in places, such as in the Jack Rabbit Slims Restaurant, and the Pawn Shop basement, later on in the film.