What is the difference between a mixtape and a LP or EP? They seem to be of similar length (some LP- ish length (10 tracks) and some EP length (3-5)).
Historically speaking, LP and EP are vinyl record formats. The LP was a full length 33 1/3 rpm album, around 45-60 minutes in length (20-30 minutes per side). The EP was developed from the 45 rpm single format (one song), it had the same format as a single, but several songs instead of one.
Mixtapes came much later, and were usually non-licensed (bootleg) compilations made by individuals using cassette tapes. (The creation and exchange of mixtapes was an important form of social interaction in the 1980's and 90's, as chronicled in the movie High Fidelity.) These were most typically one-offs made for personal use, but underground DJs --and later rappers/MCs --began duplicating and distributing them on a larger scale to showcase their skills.
In modern times, LPs and EPs are official releases. An LP is a standard length album, an EP (usually) has three to five songs.
Mixtapes are typically unofficial releases, either because they are unlicensed or because they are being released free for promotional purposes (or both --for example, Frank Ocean's Nostalgia, Ultra). Occasionally a "mixtape" will be released officially for sale --in this case, the term is being used largely to indicate it belongs in the same genre as the old street mixtapes. Mixtapes are generally full length, and can actually be longer than album length.