These are the photos of the respective Dianthus(on quite a sandy soil) and the Phlox. Can you identify the Phlox...
Off the top of my head it looks like
Phlox subulata, given the rather lax growth habit.
P. adsurgens, usually found in the form 'Wagon Wheel', does not have the needle-like foliage, and
P. douglasii (at least as I know it!) has much finer foliage and a more congested growth habit.
However, I have the impression from reading that the cultivars called "Phlox subulata" by nurserymen include hybrids with other species. To call such plants "Phlox subulata" would be a taxonomic heresy, but it's the best name we have for them.
As for not flowering, I wonder if your phlox is not getting enough sun. While
P. adsurgens is a plant of open pine woods in semi-shade,
Pp. subulata and
douglasii are definitely sun lovers.
I will say that "P. subulata" (whatever it may be) is an easy plant. The garden centres here invariably have a number of different cultivars for sale about this time of year.
About your dianthus I can offer no opinion.