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 $a \to b$ in the poset is read as encoding the information: “there is a time evolution process which takes $a$ to $b$”. 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 $x$ to the point $y$ in the manifold precisely if $y$ is in the future of $x$, i.e. precisely if there exists a smooth path from $x$ to $y$ 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 $x\darr X\darr y$ for all objects $x$ and $y$ in the poset (the category of “two-step time evolution paths” from $y$ to $x$) 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).
A causal set, or causet is a poset $X$ such that for all objects $x,y$ the interval $[x,y] = \{z \mid x \le z \le y\}$ is finite.