Learn Russian: Платная версия нравится как взрослым, так и детям. - Both adults and children like the paid version.

It translates to both ... and ... or as well as. Here, it connects the two groups who like the app: взрослым (adults) and детям (children). It emphasizes that the two distinct groups share the experience equally.

Could I just say и взрослым, и детям instead?

Yes, и взрослым, и детям also means both adults and children and is very common. However, как ..., так и ... sounds slightly more formal and strongly emphasizes the equivalence between the two groups, literally meaning just as to adults, so to children. It is often used in written reviews, journalism, or marketing.

Why are the words for adults and children in the dative case (взрослым, детям)?

This is a refresher on how the verb нравится (to be pleasing) works. In Russian, the thing being liked (платная версия) is the grammatical subject in the nominative case, while the people who like it must be in the dative case. It literally means that the paid version is pleasing to adults and to children.

Do I always need a comma before так и?

Yes. In Russian punctuation, the second half of a correlative double conjunction is always preceded by a comma.

Why does взрослым have an adjective ending while has a noun ending?