Why differential approximation ratios are not well-studied comparing to standard ones despite of their claimed benefits? There is a standard approximation theory where the approximation ratio is $\sup\frac{A}{OPT}$ (for problems with $MIN$ objectives), $A$ - the value returned by some algorithm $A$ and $OPT$ - an optimum value. And another theory, that of differential approximation where the approximation ratio is $\inf\frac{\Omega-A}{\Omega-OPT}$, $\Omega$ - the worst value of a feasible solution for the given instance. The authors of this theory claim that it has some definite advantages over classical one. For example:


it gives the same approximation ratio for such problems as Minimum vertex cover and Maximum independent set which are known to be just different realizations of the same problem;
it gives the same ratio for max and min versions of the same problem. At the same time we know in standard theory MIN TSP and MAX TSP have very different ratios.
It measures distance not only to the optimum but also to the pessimum $\Omega$. So in the case of Vertex Cover standard approximation theory says that $2$ is the best upper bound. But essentialy $2$ is the maximum ratio between the pessimum and the optimum. Thus such algorithm is guaranteed to output the solution with the worst value.


My argument pro is: in asymptotic analysis we don't take into consideration constants and low-order terms (here I recalled the quote by Avi Widgerson: "We are successful because we use the right level of abstraction.") and this is the level of abstraction for comparing algorithm's usage of resources. But when we study approximation we by some reason introduce the difference in those places where we can avoid it. 

My question is


  why the differential approximation theory so poorly studied. Or the arguments involved are not strong enough?
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.