1. 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.
Example - I gave him a gift/present on his birthday.

2. 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.
Example -
1.I gave him a pen as a present.
2.The foundation is planning to gift the land to the society.

3. The word 'gift' is also used with the inborn talent or skill that one has. It is because God is believed to have given us these things.
Ex - Picasso had the gift of painting.