The idea of a "natural moral law" doesn't come with any obvious restrictions on content built in. (So it is a favourite of religious homophobes and others with arbitrary ethical views.) But as Pogge explains:
Expressing moral demands in the natural-rights rather than natural-law idiom involves a significant narrowing of content possibilities by introducing the idea that the relevant moral demands are based on moral concern for certain subjects: rightholders.
This undermines the notion of religious duties, since God surely has no need of rights. More generally, it excludes all the arbitrary concerns that have nothing to do with any individual's interests, or harms and benefits. So that's progress of sorts. But it already excludes too much. For instance, the non-identity problem shows that one can harm humanity (or "people in general") without harming any particular person. And we might also reasonably hold that individuals have a moral duty to develop their talents, etc., whereas there's little sense to be made of the notion of a "right" against oneself.
The notion of 'human rights' is even more restricted. It is an essentially political notion, whereby the violators "must be in some sense official". Victims of theft aren't said to have had their "human rights" violated; not unless this arbitrary confiscation of property was undertaken by government agents acting in an official capacity, or some such. Human rights thus offer protection "only against violations from certain sources". (Though Pogge goes on to argue that the relevant sort of "official disrespect" may be manifested in a wider range of situations than we might at first expect -- including, for example, official inaction when protection is needed, etc.)