I don't think there is a major difference between the two words. If I had to say something, then the word 'purchase' might sound a bit less casual than the word buy, but they are mostly interchangeable. For example, here is a situation where you might want to use 'buy' instead of 'purchase' to sound less formal: My girlfriend bought (instead of purchased) a cake for me on her way home from work. And then here if you do wish to sound more formal: The company recently purchased a new point of sales system to improve customer experience.