These two Korean words have the same meaning. 아주 and 너무 both mean very or too in English. For the longest time i am always inclined to use 너무 than 아주. Last Saturday in my Korean class, my 선생님 (seonsaengnim — yes this is how teachers are addressed in Korean) gave a tip on how to differentiate the use.
According to my teacher 아주 is positive compared to 너무. The latter tends to be negative and a bit exaggerated. It may even sometimes sound sarcastic. So when complimenting people it would be safe to use 아주 instead.