Here's what I think is different about the Bad Plus, which on the face of it is simply another jazz piano trio, piano, bass, drums. 1) They play avant-garde jazz with a rock sensibility. Put another way, they play like a combination of Monk and Hendrix if Monk and Hendrix had gone through that machine in the Fly and been mixed together. 2) They have classical musician skills and they employ classical tools in their playing. Before I went to hear them play Friday night, I listened to Mozart's string quartet in in C major "Dissonance." Just happenstance, but the parallel was illuminating. 3) While they're obviously playing compositionally, they are, as is essential to jazz music in my opinion, also obviously improvising. But they're doing it in a group context. There isn't much soloing per se, and essentially none of the call and response that characterizes jazz soloing traditionally, even in a lot of free jazz. They seem to improvise together in a manner that almost appears telepathic, and while I would not go that far I would say that their solos are symbiotic. Simply put, they do group solos, as oxymoronic as that concept is.
Others have done some of these things. I think the combination, and the emphasis on group soling, is what makes the Bad Plus different.
Except for Bill Frisell, whom Bad Plus pianist Ethan Iverson credited at the concert Friday night as an inspiration. Frisell's "Unspeakable" band, which included what could be a traditional rock or jazz combo, that is a lead guitar, bass and drums, plus a violin, viola and cello, did all of the above, with somewhat more individual soloing and a lot more of the extensive inventing off base melodies that endears Sonny Rollins to me, which is probably why Bill Frisell rules in my mind as well.