> what makes it adhere and resist pulling your fingers apart?

We call that effect "tackiness". In the link a bunch of scientists (including one from 3M) discuss it better than I can.

> Also, is the pitch molecularly homogenous?

Not quite.

It is a mixture of different molecules of varying length. It's roughly the equivalent of your junk drawer full of loose cables. Yes, they are all cables, but different connectors, sometimes more than two ends, different lengths, etc.