I wasn't sure whether to ask here or go to /r/AskScience, but since this seems like it's mostly a decision made by humans to distinguish between the two, I figured I would ask here.
There's no physical separation between the two continents (to my knowledge) that would define a separation of landmass from one or the other. I know N. Am. and S. Am. aren't technically separated either, but you can clearly see the distinction between continents.
So why are Europe and Asia not considered one massive continent?
Blame/thank the Greeks. Originally the names referred to relatively small patches of land: mainland Greece, or even just Thessaly ("Europe"), and Anatolia ("Asia"). The names stuck in the Greek imagination, and expanded their scope as time went on; by the 5th century BCE, Herodotos thought of the known world as divided into three continents, Europe, Asia, and Libya (=modern Africa). More details, and references, in this older thread. Sure Europe and Asia linked up beyond the Black Sea, but that was just a bit too far away for the Greeks to throw the names away, and the tripartite distinction hung around in the Greek (and, later, western European) mind for many centuries.
Of course the names stayed like that after the geography north of the Black Sea became better known. That's probably best explained as a combination of tradition, on one hand, and on the other, a dose of eurocentrism (as per /u/lukeweiss' comment).
hi! FYI, you'll find more on this in the FAQ (link on sidebar):
It is tradition. If you look at European middle ages maps they tend to form a T within an O. Asia was on top of the T, Europe to the left and Africa to the right. Jerusalem was in the center of course as this map was made by Christians. For Christians they knew little beyond Russia and didn't bother to look. They knew you went east to get to Asia, and South to Africa. Also when you got to Asia, for the Europeans that was the middle east, there were Muslims there. Beyond the Muslims were the Indians and Chinese and probably Prestor John, the great Christian King of legend who will save all of Christendom.
Well by the 1500s they realized that it was all one big chunk of land. The Russians were pushing east and they had sailed to Japan and China by then, but by that point new ideas of race were emerging and after that in the 1900s nationalism and national pride so they just kept the division for racist and nationalistic reasons. Now it is just tradition.
Follow up question: Why is india considered a subcontinent, but Arabia (+greater syria) isnt ?
The short answer is that the distinction no longer exists in upper level scholarship. Unfortunate, it stubbornly lingers on in lower levels of education.
Why did it exist in the first place?
Simple eurocentric bias. It made for a (not so) tidy separation between white not, Christian and non, high cultured and low, etc. The border, as taught to many of us in grade school is one of the most laughable geographical thought crimes imaginable: those lowly glorified hillocks, the Ural mountains in the north, and the bosphorus strait to the south. All arbitrary silliness. But it persisted based on the perceived differences and prejudices of the European overlords. That is until post modern deconstruction came along to right the ship (ironically).