However, if you are applying for financial aids, you'd better submit your application as earlier as you could.
Let us say, if you have already handed in your application, the graduate school has prpbably got it before the winter break, before Dec. 22 or 23. Then they could forward it to your Dept as early as the very beginning of next semester, about Jan 15. Then the profs could evaluate your application then...
If you just make it before the deadline, the graduate school will hold your application for several days for gathering the stuffs, then forward it to the Dept at the end of Jan or even later.
So...
Fast~~~