They mean the same thing, except that PURCHASE is more formal than buy. :)

This question came up a day or so ago. Take a look at this: http://www.italki.com/question/263568

Purchase and Buy are essentially the same word. Purchase is a little more formal than buy. In addition, buy has usages/meanings that cannot be shared with purchase. Examples: You can use 'buy' to mean 'believe': "I told her a white lie, however I am not sure if she bought it." In regards to time we can say: "I need to buy more time!" - which usually means you're running out of time and that you're desperate for more, like to complete a project. You cannot use purchase in the two previous examples. Because of these alternate meanings, buy, is used a lot more, is a lot more common then purchase.