The A in ADT can stand for both abstract and algebraic, so it's an unhelpful abbreviation in contexts where both algebraic and abstract datatypes are relevant.

Functional programming languages tend to have good support for algebraic types (which I think you mean) but they can often do data-hiding too! In Haskell, for example, you do this by not exporting the data constructors for your type, so they can only be used within the module that defines the type.