They are close synonyms. There's little difference. They usually refer to an acid taste. The ranges of meanings overlap. With regard to food and tastes, "sour" is one of the four (or five) basic tastes: sweet, salty, sour, bitter, and perhaps "umami." One would not use "tart" here. "Sour" can be strictly factual--in the supermarket, I can buy a jar that says "sweet pickles" or one that says "sour pickles"--or it can be slightly negative. "Tart" can be a politer or more complimentary description of the same acid taste. As a description of taste, "sour" can mean "very strong" or "too strong." "Tart" can mean "not too strong" or "just right." "Don't you like this lemonade?" "No, it's too sour."--Impolite. "It's a little too tart for me."--Polite. "Do you like Red Delicious apples?" "No, they are bland and tasteless. I like a tart apple, like a Granny Smith." "Sour" can mean not just acidic, but spoiled, rancid, decayed, or putrid. "Ugh, this milk has turned sour, I need to throw it out."--Yes "Ugh, this milk has turned tart, I need to throw it out."--No, you can't use "tart" here. Figuratively, "sour" can be a synonym for anything that has gone bad. "My job was great until my company was acquired by Megacorp, and then it turned sour."

P.S. I'm a U.S. speaker. I believe that in British English, the noun "tart" is a noun--maybe a colloquialism?--for a prostitute or a promiscuous woman, or one who dresses provocatively; and similar, as a verb, "she tarts herself up."