'Fizzy' is more informal. Coca Cola is fizzy and Champagne is sparkling. There are also carbonated drinks.What are the differences between fizzy drink and sparkling drink?
There are several differences involved in the production of such drinks which somebody else might explain to you if you're still interested.
Fizzy drinks which have lost their sparkle have gone flat, as you said.
Drinks which were never intended to be fizzy are called still.