An asset is some kind of economic resource. It can be controlled or owned and it provides the owner with value, usually financial. A house is an asset. You can sell a house and get cash in return.
A currency is a system of money, a means of exchange. You used currency to buy a latte this morning, or so you could watch the latest season of Altered Carbon on Netflix
If its hard to tell the difference between the two, consider this. Assets tend to have a built-in promise of increased future value. A house tends to increase in value in time and thus, is an asset. Its not very liquid, unlike cash, and therefore cant really be thought of as currency.