Differential forms or tensors for Modern Theoretical Physics?

Question

Are theoretical physicists nowadays adopting the approach of tensors or differential forms for their work in higher dimensional manifolds? Or are both approaches used together? Frankel, in his book, Geometry of Physics, emphasizes that there are advantages to both depending on the problem at hand. 

Are more physicists learning differential forms in their work?

Can you recommend a good textbook for learning differential geometry as practiced bya modern physicist?