This question comes up all the time! Here's the answer I usually give:
In the example you've given, there is no difference. A general rule is that all presents are gifts, but not all gifts are presents. A present is something which you choose specially to give to a particular person, usually on a special occasion, such as their birthday or their wedding. This can also be called a gift.
However, there are many situations where gifts are given which are not presents. For example, a company could offer gifts to its loyal clients. A rich person might give a work of art to his city as a gift. Magazines sometimes include free gifts for their readers. These are not presents, but they are gifts. As you can see, 'gift' has a broader meaning than 'present'.
One final note:
You ask a lot of questions and you get plenty of answers, but you never seem to thank people for helping you with your English. Don't you think it would be courteous to acknowledge the replies which you receive? Just saying...