I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given choice of invariant polynomial, and the construction of Chern-Simons forms is a functor on the category of principal bundles with connection.

So somehow this construction has to do with differential cohomology and differential characters. Admittedly, I only have a very vague understanding of differential characters and differential cohomology. I have skimmed through the Cheeger-Simons paper but the big idea hasn't popped out at me yet.

Also, I am interested in invariants of principal bundles with connection, but from the nlab page on the topic it seems that differential cohomology has to do with circle bundles with connection. Is this indeed the case? Or is it useful for principal bundles with other sorts of structure groups?

Essentially, I'm hoping to see the big-picture before I invest myself in learning differential cohomology and differential characters.