Depends on what exactly you mean by "important", I'd say. And then there's the thing that Europe around the Med is rather a bit different in many ways from the Europe north of those mountain ranges ringing it...

However, this discussion is about the contemporary perception of where the division between "Europe" and "Asia" went - and I'm pretty sure the only folks in Europe that cared about the matter were the around the Med, and even there probably only near the actual border zone. Your average Briton, Gaul or German could hardly have cared less, after all, and I rather doubt if the question was particularly central to the average Latin either.