Primarily one of duration and subject matter. Personally, I think all this differentiation is silly, but anyway, there are three types. A press gaggle takes place anywhere other than in the formal setting of the WH media room, and is called on relatively short notice. Typically, gaggles occur on AF1 while the president is travelling.
A press availability is when some dignitary is visiting with the president. After private discussions, both will make themselves available for a few questions from the media. (Usually jointly, but not always.) Usually each leader will answer only two or three questions at these availabilties, rather than the more wide-ranging questions asked at formal press conferences.
The latter are just that: formal conferences in which the president (on occasion, jointly with an A-list leader like Tony Blair) takes a wide range of questions from the media over a relatively long period of time -- as much as 45 minutes to an hour, but rarely longer.