Optimization problems with same optimal value, but different approximation behavior
What about something like minimum vertex cover vs. minimum fractional vertex cover (i.e., optimal solution of the LP relaxation of the vertex cover problem) in bipartite graphs?

These two problems have the same optimal value.

However, if I give you a black-box algorithm that finds a $(1+\epsilon)$-approximation of the minimum fractional vertex cover, this does not help much if you would like to output a $(1+\epsilon)$-approximation of the minimum vertex cover.

This can be formalized in some sufficiently weak models of computation. For example, if we look at distributed algorithms, one can show that a $(1+\epsilon)$-approximation of the minimum fractional vertex cover is easy to find, while a $(1+\epsilon)$-approximation of the minimum vertex cover is hard to find in bipartite graphs. (Here "easy" is e.g. constant time in the LOCAL model and "hard" is e.g. at least logarithmic time in the same model, assuming a sufficiently small constant $\epsilon &gt; 0$ and some constant upper bound on the maximum degree. So there cannot be e.g. constant-time approximation-preserving LOCAL model reductions between the two problems, even if the optimum is the same.)