For me, "snooze" is a synomym for "sleep" while "doze" is light sleeping.
If I'm snoozing away, I may not hear the door open, but if I'm only dozing, I probably will. I tend to doze in a plane, but not really sleep soundly.
"Snooze" is a special function on an alarm clock that turns off the alarm and sets it to go off again in a few minutes. In college, I used to set my alarm early and hit the snooze button 4 or 5 times. If you "snooze" like that, I would not call that "dozing."
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