In the 20th century, classical music went a lot of interesting ways. A lot of the really innovative composers were focusing on dissonance and atonality — Bartók, Stostakovich, Varèse, Stravinsky. With that, there starts to be a challenge to find music that grasps you instantly. I tend to find that once I figure out
something about a composer, his works start making sense to be, but before then it frequently sounds like noise.
When I think about modern composed music, my utterly-amateur mind goes in three directions — jazz, minimalism, and scores. There's a lot of interesting music being made for movies — do the scores of Jaws and Indiana Jones or Sweeney Todd and Phantom of the Opera count as classical music? Certainly musically there are certain things in common. Jazz, meanwhile, is an absolute paradise of talented musicians and musical experimentation. Charles Mingus comes to mind, as does John Zorn as a more modern composer.
The only movement I'm really aware of in modern classical music that has a distinct classical taste is that of minimalism. Philip Glass is an obvious first name, but then Steve Reich, Terry Riley, and La Monte Young were all composers whose music emphasized certain kinds of repetition. But when you begin listening to them, you realize how diverse each one is and how it's almost silly to group them together. Riley's music tends toward electronica and unpredictability, Glass eschews the term altogether, and Young's music tends towards categories like "drone", at which point you're talking a genre that includes a lot of rock music, also. Then you look at how Riley influenced bands like the Who (who named a song after him) and how La Monte Young worked with John Cale of the Velvet Underground, and it starts looking like that branch of classical music got submerged in avante garde rock.
But that's all the perspective of somebody who admittedly knows less than he wants to. If the classical music scene is different than that, it's done a damn good job of hiding it: I've found no other category of music that might help me understand what's going on today (and I'd appreciate the helping hand of any user who knows more).