Which is the better music school?
Depends on what instrument and type of music you want to pursue.
Very different schools. Juilliard is very very very selective.
Pet peeve. Correct spelling: Juilliard. Note the “i” following the “u”; two in total, one in each syllable. Auto-correct often “corrects” wrongly, dropping the first “i”. Don’t believe it!
Juilliard the best for classical. Berklee for contemporary.