The OALD gives this usage note:
The adverbs deep and deeply can both mean ‘a long way down or into something’. Deep can only mean this and is more common than deeply in this sense. It is usually followed by a word like into or below:We decided to go deeper into the jungle.
Deeply usually means ‘very much’:deeply in love ◇ deeply shocked. You can use deep down (but not deeply) to talk about a person’s real nature:She can seem stern, but deep down she’s a very kind person. ◇ She can seem stern, but deeply she’s a very kind person.