These two words do not always have the same meaning.
A present is something which you have specifically chosen for a particular person, often for a special occasion, such as a birthday present. It is possible to also use 'gift' in this context.
However, there are many situations where you can use 'gift' but not 'present', particularly where the gifts are not specific to the recipient. For example, companies often try to attract clients by handing out free gifts (not presents) . A toy wrapped in cellophane on the front of a kid's comic is a gift ( not a present). A sum of money can be a gift (not a present). Visiting diplomats often bring gifts to their host countries (not presents).