Hello Marina, The two words are used most of the time interchangeably 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. However there is a slight distinction between the 2 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 . "She got many presents on her birthday from relatives and friends." " He made a gift of million dollars to his old university." The word "gift" is also means the inborn talent or skill that one has. " Picasso had the gift of painting." In this case the word 'present' is not a synonym nor has a close meaning to the word "gift".

To add to Cherry said, there is also a legal distinction. You could give a gift to a charitable organization. The word gift here would refer either to money or some physical item, like a car. You would not say that you gave a present to a charity.