Herodotus wrote about the border but said he didn't even know why the continents divided this territory — a territory that was obviously one unitary land mass, and to the ancient Greeks, there really wasn't any meaning to Europe and Asia. They were just purely geographical terms.
Of course today, that's no longer the case. When you say Europe, all sorts of cultural, political, and geopolitical associations come along with that — even more so when you talk about the West, which people often use interchangeably with Europe.
Associations with "the West" include things like civilization, enlightenment, democracy, individualism, and materialism — while "Asia" or "the East" has been associated with the opposite of all those things.
The thing that's most interesting to me is that these countries along this artificial, geographical border are exactly those where you have some of the most profound and serious debates about national identity: Where do we fit? Are we European? Are we Western? Are we Eastern?
Plenty more to come...