As with all labels and terms, these words can often mean slightly different things to different people. As a general rule, however, a "bottom" is usually someone who only takes on the submissive role within a play session or scene for a limited and previously negotiated time period. During the scene, they inhabit the role of the bottom, receiving the actions of their partner, who would in this situation would be referred to as the "top."
A "submissive," on the other hand, tends to be someone that embodies a role of submission within a relationship dynamic, either full-time or part-time, depending on what has been negotiated between the people involved. Unlike a bottom, who mainly plays a receiving role in sexual interactions and scenes, a submissive may approach their label as signifying a more psychologically complex, 24/7 commitment. In exploring a d/s lifestyle, a submissive may agree upon certain rules and structures, with their dominant partner(s). Submissive and dominant roles can, of course, be inhabited by the same person, although it is quite common for one to prefer a particular role over the other.
As with all labels and terminology, these are only fairly broad definitions and everyone has their own (slightly different) take on what these words mean to them. Therefore I always recommend that you take time to talk to those you are interested in exploring these terms with, to find out where they stand and what "bottom" or "sub" means to them.