First in general, neons are not active fish, nor boisterous, but sedate and quiet. However, once they are settled in the aquarium (they may take several days to settle to a new home) they should come out more than what you describe. So that means they are likely frightened of something.
Water conditions, the aquascape, and other fish species can all influence a fish's stress level. Fish need hiding spots, or they will be stressed. The fish doesn't know it is safe, only that it is vulnerable. The natural habitat of a species is the key to the sort of aquascape it requires to be healthy. Plants can help, as can chunks of wood, or even artificial decor. Provided the fish is not forced into the open all the time, it should be OK once it settles.
The above pertains to both posts 1 and 2. Now to post 1, the Black Skirt Tetra may well be the problem. This species is feisty, active, and often beligerent and harassing for a tetra. I would remove this species. The platy is not frightened of the Black Skirts because platies are fairly active and not normally shy or timid themselves.
Byron.