As nouns, they are synonyms and can generally be used in any situation. Perhaps the only situation that "gift" is used not "present" is that a charitable donation is always called a gift. As verbs, to gift and to present mean completely different things. To gift means to give something to someone as a gift or present, meaning for free. To present means to give or offer something and it has multiple uses. I present you with the bill. I presented my report.

A gift is more formal and usually applies to something that is worth more. However, you could use one for the other and it would be correct grammatically. He gave is daughter a doll as a present for her birthday. The staff got together and bought Donna a present. The Johnsons made a gift of their rare books to the library. Note that in the first two, you could have said "gift". You could say "present" in the third, but it would sound just a little out of place - but not much. Also, if someone has a talent, we say they have a gift.

Yesterday is history. Tomorrow is a mystery. And today? Today is a gift. That's why we call it the present.

They're the same word.

Thanks for the explanations :)