Why differential approximation ratios are not well-studied comparing to standard ones despite of their claimed benefits?
I am not familiar with the notion of differential approximation, and I do not have any theory why it is not well-studied.  However, I would like to point out that it is not always desirable to describe the performance of an approximation algorithm by a single measure.  In this sense, I find it difficult to agree that some measure is better than another.

For example, as you stated, minimum vertex cover admits a polynomial-time 2-approximation algorithm whereas it is NP-hard to approximate maximum independent set to any constant ratio.  Although I understand that this can be surprising at first sight, it has a legitimate meaning: (1) minimum vertex cover can be approximated well when it is small but (2) it cannot be approximated well when it is large.  When we state that it is NP-hard to approximate minimum vertex cover (and maximum independent set) with any positive constant differential approximation ratio, we are effectively ignoring the property (1).  It is probably good enough for some purposes to ignore the property (1), but certainly it is not always the case.

Note that we do not always use approximation ratio to describe the performance of approximation algorithms.  For example, to state an inapproximability result based on the PCP theorem in its full generality, we need the formulation based on gap problems.  See my answer to another question for details.  In this case, neither using the standard approximation ratio nor using the differential approximation ratio allows us to state the result in the full generality.