I'm not going to give you a whole history lesson here, but it started in the late fifties. The first wave is actually the forefather of reggae. It's changed a lot since then. The second-wave ska bands were the Specials, English Beat, and Madness. Now the third wave is a lot more diverse.
Some people refer to it as the same thing.
There's a difference, but I guess it's not really all that important, as long as you understand that some bands that are still considered third wave ska bands have more of a traditional sound. Others have maybe more of a two-tone sound. Ska-core usually combines ska with punk rock, or pop, or other styles of music, and that's what we do. Shari - Ska lyrics are characteristically goofy and fun. Do you think that's a correct description in general.