In maya you don't really animate the individual vertices for something like this, you can animate vertices using deformers like the 'lattice' or making 'blendshapes'.

For the example you gave with the paper, it would be best to line up the vert's, can snap one set of vert's to the other set, aslong as they share the same position in 3D space then you won't be able to see the gap between them.

Ask if you need some more help on the idea.

Chris

PS. I shall mock up something quickly like your demo to show the idea, will probably use a lattice or bend deformer.