A causet or causal set is essentially the same thing as a poset, more precisely a locally finite poset.
The difference in terminology indicates the restriction to certain applications: regarding a poset as a causal set means to regard it as a model for something like a space with time evolution: a morphism in the poset is read as encoding the information: “there is a time evolution process which takes to ”. Equipped with this interpretation as causal sets, posets find applications in
relativistic physics, where every globally hyperbolic Lorentian metric on a manifold equips that manifold naturally with the structure of a causal set: there is a morphism from the point to the point in the manifold precisely if is in the future of , i.e. precisely if there exists a smooth path from to whose tangent vector is everywhere non-spacelike with respect to the Lorentzian metric. Moreover, the Lorentzian metric on a manifold can essentially (need to dig out the details here, see discussion at smooth Lorentzian space) be reconstructed from this poset structure and from a measure. This has led to some attempts to use posets as a foundational concept for relativistic physics.
For more see smooth Lorentzian space
The only technical distinction between the notion of posets and that of causal sets is that for a causal set the under-over category for all objects and in the poset (the category of “two-step time evolution paths” from to ) is required to be finite. This means these are required to have a finite set of objects (and hence necessarily, being a poset, a finite set of morphisms).