Easier and Harder instances of NP-complete problems? Following previous questions here and here around $NP$-complete problems (and P vs NP).

Are there "easier" or "harder" (sets of) instances of an $NP$-complete problem?

If yes (which i assume), how a polynomial-time mapping between $NP$-complete problems maps between easier or harder (sets of) instances?

A conjecture  (related to directions of previous questions) is that there should be considerable mapping betwen "easier" to "harder" and "harder" to "easier" (if i may, with probability almost 1).

PS. This in effect tries to study (a little more in depth) the role of polynomial-time mappings used in $NP$-complete reductions.

Thankx
If P=NP, then all NP-complete problems are equally difficult from the perspective of polynomial-time reductions.

On the other hand, if P!=NP, then we have ways to differentiate NP-complete problems, for example:


Approximable problems - NP-complete problems where we can find a polynomial-time algorithm that gives us almost the desired answer (within a constant factor).  For example, there is a trivial algorithm to produce a Vertex Cover within a factor of $2$ of optimal, although the optimal answer is NP-complete.
Somewhat approximable problems - NP-complete problems that can be approximated, but not within a constant factor.  The classic example is the set cover problem, which can be approximated within a fraction of roughly $ln n$ for $n$ sets, but cannot be approximated with a less than logarithmic fraction (unless P=NP).
Unapproximable problems - NP-complete problems for which (under accepted assumptions) no polynomial-time algorithm exists that approximates the solution within any polynomial factor.  An example is the max-clique problem.
Fixed-parameter tractable - problems that run in polynomial time if you fix one parameter.  The classic example again is vertex-cover, which can be computed in time $O(2^k n)$ for a vertex cover of size $k$ in a graph of size $n$.


In short, under accepted assumptions, there are varying difficulties of NP-complete problems.