Discuss and learn French: French vocabulary, French grammar, French culture etc.
I'd better say l'avenir but both are correct.
Everything has a "futur" but not necessarily an "avenir".
You can't say "Cette histoire d'amour n'a pas de futur" that sounds very weird... But you can say "Cette histoire d'amour n'a pas d'avenir" instead. "Futur" is only a chronological period of time whereas "avenir" is a period of lifetime to be fulfilled and constructed.
Both are correct but the point of view is not the same.