Money is anything used to exchange value in transactions.

Suppose I have a 1985 Corvette and you have chickens.  Also suppose you'd like to have my car.  I may not particularly need chickens, but if I know that pretty much everyone else wants chickens, we will be able to agree on a number of chickens to trade for my car.  I then trade other people chickens for things that I want from other people.  The chickens here are acting as money here without being currency.

Currency is a specific kind of money, often called fiat money, that is established by governments to centralize an economy onto one kind of transaction medium.  In most, if not all, systems currency takes the form of coins and paper bills, though this is quickly being reduced to balances stored and dispersed on credit/debit cards.

In common use, though, the two are more-or-less synonymous.