This is probably for the bigger anime fans.
Anime News Network has a poll up today, apparently it's been a pretty heavily debated topic for a while.
Both of these ladies are probably 2 of the best when it comes to musical composition in anime. To point out of their more popular works for casual anime viewers:
Yoko Kanno is the lady behind the authentic jazz of Cowboy Bebop and the epic orchestration of Escaflowne.
Yuki Kajiura is responsible for the music of the last 2 Xenosaga games and the .hack//* anime.
People who are deeper into anime should know the other works they've done, so I won't list more.
They are both awesome and their soundtracks are usually great, even if the particular anime they are working on isn't so hot. However, my vote goes to Yoko Kanno, because her range is so diverse. To go from Orchestral/Choral Escaflowne to Jazzy Bebop then to the very modern Industrial style of Ghost in the Shell, all with outstanding soundtracks, is a range that Kajiura has yet to even attempt.
While Kajiura's stuff is awesome and often beautiful, I find that .hack, Noir, Madlax and Tsubasa kind of sound the same. Kind of like how John Williams' music, which is very good, sounds very similar. The melodies themselves may be different, but listening through Star Wars, Superman and Harry Potters' soundtracks you'll hear a lot of similarities.
It's great that they have a certain style, and I'm still a big fan of theirs, but this vote goes to Yoko Kanno.
It'd be interesting to see who some of the other big anime fans there are on this site.