The words gift and present are often interchangeable, as they both mean almost the same thing. They both suggest that something is given to someone without the person who gives it wanting anything in return.
There are, however, subtle differences. The word present is applied in the sense of donation, while the word gift is used in the sense of reward. Giving something to a family member or close friend is more casual than giving the present in a formal situation.
In a formal situation, if you are giving something to someone, then it is a gift, not a present. The present demonstrates your appreciation for that person as an individual, while a gift is something that might be impersonal, but it has some value to it.