In daily colloquial use they are the same, but gift does sometimes have a slightly different meaning than present. Officials of countries officially give gifts, to other officials, never presents. Presents are mostly given to children for birthdays etc but you can give them a gift also. At a wedding you give either a gift or a present they are interchangeable. Presents are what you receive without asking for them usually at a special occasion or by someone to show appreciation. Gifts are given to charities and to a countries officials etc as a gesture of welcoming. Which means they are interchangeable except when you give to charity it is a gift and when something is given for official state and country visits by statesman ambassadors etc.

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...

Also I thought it would be good to point out that the words themselves have other meanings than the ones in your sentences. Gift is pretty simple, it has a meaning as a verb to give a gift. But present has several different meanings that are much broader, from the present time to presenting, or introducing someone, to present with symptoms of an illness, and several more.