Context: related to this answer.
I would like to see an example to emphasize that approximation behavior depends not only on the optimal value but also the set of solutions. This makes sense logically. If two problems are duals of each other, then they have the same optimal value but can have different behavior with respect to approximation (eg: independent set and vertex cover). But, is the same possible if both are minimization problems, or both are maximization problems?
Are there two minimization problems with same optimal value, but different behavior with respect to approximation?
Existence of such a pair of problems justifies the definition of approximation factor preserving reduction.