Formality. Purchase is more formal. More adquirir than comprar.

Also purchase can be used as a verb and a noun! Example: I'm going to purchase that car that we saw yesterday. (Used as a verb) I made a purchase yesterday. (Used as a noun) Are you ready to make a purchase, sir? (Used as a noun) Whereas, buy you can only use as a verb. Example: I'm going to buy that car that we saw yesterday. (Used as a verb) I made a buy yesterday. (INCORRECT)

There is no difference in meaning. "Buy" is the simple, everyday word. "Purchase" is the more formal word. I might tell my wife "I'm going to buy a new tablet." When I get to the store I would say to the salesperson, "I want to buy a new tablet." I would not say "I want to purchase a laptop." When I open the box, the registration card from the manufacturer would say "Congratulations on your purchase of a fine QXZ product! You can look forward to many years of reliable service from it. In order to provide you with the optimum customer experience, we request that you take just a few moments to register your product with us." The department of a business that buys things for the business is called "purchasing."