The distinction between the two is to some extent arbitrary. The Urals are considered to be the border between Asia and Europe. Regions such as the Caucasus can be counted as part of either and some countries such as Russia and Turkey are partly in both, with Turkey's largest city Istanbul being the only city to straddle two continents. There are some other locations where the border is clearly marked, such as Europe & Asia border monument in Yekaterinburg.
Of course the two have always been linked; see Silk Road for the historic trade routes or Trans-Siberian Railway for probably the most famous overland trip between the two continents today.