"Purchase" is used in more formal settings, like business transactions. You can't really use it in informal conversations, like, "Come, I will purchase you a beer" or "Mom, could you purchase me a guitar?" Actually, you would have to say "Mom, could you purchase a guitar for me?" but it sounds rather awkward.

There is no difference between buy and purchase. They are synonyms. I would say that purchase sounds more sophisticated than buy, but both mean the same.

They have the same meaning.

buy- informal purchase-formal agree with Viviane