They are more or less synonyms, but I think there is a difference. There's something slightly more formal - and impersonal - about the word 'gift', at least to the British ear. For example, if you buy a product, the company might give you a 'free gift' along with your purchase. This is something quite detached and impersonal, and you would never call this a 'present', because it's not a real present - it's just a marketing trick. A present is something more personal and genuine, such as you might buy your friend for their birthday.

"Many of us tend to use the two words interchangeably in most contexts. Of the two, present is the more informal. The two words are used with things given to other people without expectation of return or compensation. It is not uncommon to hear people say, I gave him a gift/present on his birthday. Both the sentences are grammatically acceptable. Careful users of the language make a distinction between the two words. A Gift tends to be much more valuable than a present. It usually passes from the rich to the poor, from the high to the low. A present, on the other hand, passes between equals or from the inferior to the superior. • The children brought a present for the Minister. • I gave him a pen as a present. • The foundation is planning to gift the land to the society. • He made a gift of two million dollars to his old university. "

I think it depends on people

They are pretty much synonyms for each other, although I would say that "gift" is slightly more of an American term.