Both words describe an emotion or belief that is felt to a great extent, but there are also differences. Examples: I'm deeply (not strongly) grateful to you. He was deeply (not strongly) hurt by her accusations. He could deeply (not strongly) feel the support from his family (Here, "deeply" is used to express the depths of one's sadness or gratitude in response to someone's help or kindness.) They strongly (not deeply) support corporate transparency. I feel strongly (not deeply) about this. (Here, "strongly" is used to express a serious and firm conviction of one's principles.)