The Size Hierarchy of a Landmass
Caveat: I'm answering this in a popular context. There may be more technical definitions, which I'd be interested in hearing about.

It seems to be generally accepted that continents are not, themselves, islands. This is a common confusion in pub quiz questions, where people argue over whether the largest island in the world is Greenland, Australia, or (if somebody especially pedantic is around) the combined landmass of Europe, Asia and Africa.

So the question comes down to "what is a continent"? It's more of a political / historical / geographical unit than a topological one. If continents were demarcated by coastlines, then Europe, Africa and Asia would all be one. Additionally, many small mid-ocean islands, which are not on continental shelves, would end up being continents.
It's tempting to think of continents as relating to tectonic plates, but even if we disregard mid-continent faults and "complicated" areas, Europe and Asia are hard to separate.... and the 6 continents as currently thought of were pretty much set before plate tectonics was advanced as a theory. Wikipedia has an interesting article on historic boundaries between continents; the one between Europe and Asia is especially ambiguous.

I'm not aware of any definition of "subcontinent", and have never heard this term used for anything other than the area around India, so I suspect this is also a political and historical term.

So to answer the question: No, there is not a size-based distinction between islands and continents, although the smallest continent is indeed bigger than the largest island. The definition of an island would seem to be "a landmass surrounded by water that is not thought of as a continent".

As for "islet", again, I'm not aware of any formal definition. However, the UN Convention on the Law of the Sea does distinguish between islands and "rocks", where the latter are islets that are too small to be habitable. This is an important definition, because unlike islands, rocks are not considered when defining a nation's exclusive economic zone (the area in which they may control who fishes, drills for oil, and so forth).