Hmmm...I would say there isn't much difference at all. Purchase sounds to me a little more formal than buy. The formality is more related to the person speaking than the action. So if someone says "I purchased a new car." instead of "I bought a new car.", it sounds to me that the person would be more "correct" or formal. It doesn't make a difference to the car if it was purchased or bought, it's the same thing.

There is no difference in meaning, and - once again - it's a question of formality. British English uses the word 'purchase' even less than American English does, and it would sound odd to use the word in everyday conversation. You would never comment to a friend, for example, that you had 'purchased' something - you'd always say 'bought' or the all-purpose 'got'. However, if you were writing a letter of complaint to a company about a faulty product, it would be appropriate to use 'purchase'. By the way, the noun 'a purchase' is quite useful, as there isn't an equivalent noun from the verb 'to buy'.

They're very similar! To me, "buy" is less formal. It conveys the idea that you got something. "Purchase" more strongly implies the actual exchange of money for something. Most Americans will use "buy" in everyday speech, though :)