In many cases, they can be used interchangeably.
I'm sorry we have to deny your request/refuse your request.
If you're using it before an infinitve, use refuse: I refuse to discuss it any further. I refuse to consider it.
If you use it before a noun:
If I refuse, someone is offering it to me, but I won't take it. I refused the money she tried to give me.
If I deny, the word that comes next is the person that I won't let have [whatever it is that's being denied]. I denied you [the satisfaction of seeing me cry.]