There is a difference in meaning, so the “right” form depends on what you want to say.
Between indicate that something happens involving (at least) two specific (types) of entities.
A feasibility survey has now been completed in India to establish a network to felicitate contacts between small and medium enterprises.
This sentence means that contacts are facilitated between small enterprises and medium enterprises.
While it is possible that this is the intended meaning, I doubt it. It would make sense if one would wrote something like:
We are trying to facilitate better contacts between citizens and the government.
Usually, the expression small and medium enterprises is used to refer to all those enterprises that are not considered “big”. So we are talking about one group of enterprises.
Among is used to indicate interaction between members of one specific group(*). So it is very likely that the actually intended meaning of the sentence is indeed:
A feasibility survey has now been completed in India to establish a network to felicitate contacts among small and medium enterprises.
This would mean that enterprises in the groups of small and medium enterprises form contacts with other enterprises in the same group. To my mind, this makes more sense than the version with between.
Another simple example of this use, to contrast it with the contacts between citizens and the government:
We also try to facilitate contacts among citizens.
It can also mean the existence of something within a group, but that meaning doesn't seem relevant here: I wanted to find the strongest men among the athletes