How does a co-NP problem differ from an NP (its complement) one?
A co-NP problem differs from its complementary NP problem exactly by being its complement! That's the whole story.


  NP problems should accept YES instances -> instances where the answer is yes.


I think you've confused yourself. Every problem, of any complexity, is a division of all possible instances into "yes" instances and "no" instances. For TSP, the "yes" instances are the maps where the salesperson can visit all the cities within the distance requirement and for co-TSP, the "yes" instances are the maps where the salesperson can't (i.e., the "no" instances of TSP).

What differs between complexity classes is the amount of resources and model of computation that are required to tell the difference between a "yes" intance and a "no" instance. For NP, we use a nondeterministic Turing machine running for a polynomial number of steps and say that the machine accepts if at least one computation path accepts, and rejects otherwise. For co-NP, we do the opposite: the machine rejects if at least one computation path rejects, and accepts otherwise.