There's no difference in meaning when they're used as verbs, but "purchase" is more formal. You'll hear "purchase" more often in the business world and "buy" more often in everyday life. As nouns, a "purchase" describes the transaction when something is bought and "buy" means a "bargain" or a "good deal."

I'm going to buy a meal. I'm going to purchase a meal. Buy is more commonly used in everyday speech. People talking amongst each other. I'm going to buy it. Look at what I bought. Who would buy that? Purchase seems to be used more in writing legal terms or conditions. Proof of purchase needed for all returns. Limit one per purchase. All purchases are final.

As verbs, they have the same meaning. However, as nouns their meanings differ. A "purchase" is something you have bought (or purchased). A "buy" is something that is offered for sale at a very attractive (low) price.

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