They are synonyms. Just pay attention to the gender - děvče, even though it means a girl, is neuter.
This is saying that "jiné děvče" translates to "another girl." I was under the impression that "další" means "another/additional" and "jiné" means "different", and that jiné and další were not interchangeable. Can somebody help clarify this please?
There is some overlap. Depends on the context. Jiné and další can be sometimes interchangeable although they are not the same.
It might help to think asking for another cookie is not the same as asking for a different cookie.
I have it arranged in my head this way: While both jiný and další can be translated as "other/another," jiný carries the sense of "different" but další carries the sense of "additional." If context is available, it is usually clear which is meant. For this sentence, where no context is given, both are accepted.