What's the different between:
When they're all supposed to mean "to pay money"?
They don't all mean "to pay money" - in fact none of them mean just that.
"Buy" means "to pay money and obtain something" i.e. you pay money in exchange for the product or service. This is the verb form. In noun form it means the item that was bought.
"Purchase" is a synonym for buy and means the same when used as a verb and noun.
"Buy up" means to buy everything available in the store.
"Buydown" is a financial term used to mean paying off some part of a loan and reducing interest rates.
Buy-Up - investor's lingo http://www.investopedia.com/terms/b/buy_up.asp
but may also refer to buying everything of a given item
Buy-down - mortgage lingo http://en.wikipedia.org/wiki/Buydown
Purchase Aquire something for money - synonym of Buy
The difference is "buy down" which is a :
Edit: "Buy up" has a nuance that "buy" and "purchase" doesn't have, and that is: