Sometimes, the difference between similar verb-forms is that one needs an object and the other doesn't.
"Recover" doesn't need an object. However, with "get over", our question is, "Get over what?"
"It took me time to recover from the illness." (note that we add a preposition: from)
"It took me time to get over the illness."
"Get over" is also in the common/colloquial register, whereas "recover" is more formal.