Everybody whines about the word "emo". Get over it. Very obviously there is "old emo" (ex: Jawbreaker) and "new emo" (ex: My Chemical Romance, every other horrible band my old roommate listened to).
Don't like the word? Fine. But popular use dictates the meaning of words, not you. (this is not aimed directly at you, zanzibar, I just hear that stupid argument way too much)
On Topic:
Shot Baker
New Mexican Disaster Squad
Dinosaur Jr.
The Vacancies
[indie kid snob mode turned on]
Jawbreaker is modern compared to most of the other ones I listed, and I wouldn't call them emo. I'd call them punk, or melodic punk or pop punk, before calling them emo. The thing is that they were heavily influenced by Husker Du who were definately an emo band. Further, Further Seems Forever was definately influenced by Jawbreaker and they led to Dashboard Confessional which is considered an emo band (pop rock IMO). Also, Jawbreaker was a big influence to the wave of bands that included mineral, alkaline trio, yellowcard, rufio, midtown, and many others - and those bands are usually called emo or punk emo. Rites of Spring and Husker Du were connected to the same scene and Rites of Spring led to Fugazi which led to countless other bands and all of them get called emo. [indie kid snob mode turned off] But barely any of them sound like one another!
So emo is some sort of Punk offspring?
Yes and no. In the 80's and early 90's, "emo" was used to refer to a particular offshoot of hardcore. At some point in the 90's, reviewers and kids trying to be hip started using it in order to appear like they were in the know. So now it's used to describe everything from alexis on fire to bright eyes to weezer to jawbreaker to penfold to built to spill.