In Probabilistic Graphical Models, are Cliques and Clusters the same?
A clique is a rigorously defined, exact part of a graph $G=(V,E)$; 

$\qquad\displaystyle C \subseteq V \text{ is clique} \iff \{ \{u,v\} \mid u,v \in C, u \neq v \} \subseteq E$.

A cluster is more general, but also more nebulous. Here's what Wikipedia has to say:


  Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in other groups (clusters).


So, a cluster also is a set of nodes that is "dense" in some sense, depending on the metric used. However, while is it always clear whether adding a note increases the size of a clique, it's not always clear from looking only at one cluster whether adding a node to it is better; the quality of a clustering is defined on the whole graph.

As the Wikipedia article shows, there are many notions of clustering. Cliques can be seen as one of them.