Dat is (currently) focused on synchronizing files in a peer-to-peer network.
Noms can store files, but it is much more focused on structured data. You put individual values (numbers, strings, rows, structs, etc) into noms, using a type system that noms defines, and this allows you to query, diff, and efficiently update that data.
Also Noms isn't peer-to-peer (although we hypothesize that it could run reasonably on top of an existing network like IPFS).
dat has changed so much, and there has been so much hype and tooling (probably now broken) around it, and yet it doesn't seem to be delivering anything, nor there seems to be many data willing to be published with it.
Could you tell us a bit about how this compares to dat? http://dat-data.com/