I am posting this as a community wiki post because I am just reiterating things said in the comments, but I think you are missing some basic points and so wanted to highlight those.

You should read the Wikipedia article Close-packing of equal spheres.

In particular, you should look at this image from that Wikipedia page:
It explains how to iteratively create a close packing of spheres (one attaining the maximum density) in layers. In the first step, we follow the "A" pattern, in the second step, we follow the "B" pattern, and then in the 3rd step we have a choice of either "A" or "C" pattern. We can continue in this way as long as at each step we choose a different letter than the one we are at. So different sequences of A's, B's, C's give different packings.

The cubic close packing corresponds to the pattern ABC ABC ABC ..., while the hexagonal close packing corresponds to the pattern AB AB AB .... Taking some other pattern, but which still is periodic, like say ABC AB ABC AB ..., will give another close packing of spheres which is not either of those two, but which is still "triply periodic" in the sense of your postpostscript. So the literal answer to the question in your postpostscript is: no, there are other packings beyond those two which are triply periodic and just as efficient.

**EDIT**: Packings of these kind are called Barlow packings. As suggested in the comments, perhaps your question ought to be whether all periodic packings of maximum density are Barlow packings. I am fairly certain this must be an open question. Again, referencing something brought up in the comments, in Kuperberg's "Notions of denseness", he says "It seems likely that a sphere
packing is weakly recurrent among dense packings if and only if it is Barlow, but not all Barlow packings are uniformly recurrent." Here "weakly recurrent" and "uniformly recurrent" are technical notions having to deal with the issue of asymptotically-zero-density deformations, but morally they should be similar to periodic.

On the other hand, if maybe your interest is instead to single out the cubic close packing and hexagonal close packing among all maximally dense sphere packings, then you may be interested in this quote from "Recent Progress in Sphere Packing" by Conway, Goodman-Strauss, and Sloane: "Only two of these [Barlow packings] are 'uniform,' however, in the sense that there is a symmetry of the packing taking any one sphere to any other." So possibly that could be a characterization: the CCP and HCP are the only maximally dense sphere packings where the symmetry group acts transitively on the spheres. I am not sure if that is known.

lattice packing, the centers of the spheres form a discrete subgroup of $\mathbb{R}^3$. Aperiodic packingis the union of finitely many translations of a lattice packing, and I believe that this is what you mean when you say a "triply-periodic" packing. ABarlow packingis obtained by stacking hexagonally packed layers, where at each layer you pick one of the two possibilities. $\endgroup$20more comments