Is there any other difference between the cakes and the cup cakes besides the size? I don't have the small cups, so I place the cup cake batter in big vessels, and I think I have prepared cakes.

Now, I'll get the small cups and place the same batter in the cups, and call them cup cakes. 

Does this make sense? Or I am missing a point?
You are correct: Cupcakes are just regular sponge cakes cooked in approximately cup-sized tins.