What is the difference between a clique and a simplex? I have seen several descriptions of simplicial complexes and clique complexes as being a combination of simplexes and cliques.
I have heard descriptions of cliques being a subnetwork or subgraph wherein all nodes/vertices are 'connected' to each other.  I have also heard similar descriptions of simplices or definitions of the simplex.
Are these the synonyms, or is there a subtle difference between the two?
For example, is a simplex simply a subgraph wherein there are no unconnected vertices, but not strictly requiring all-to-all connections? Does the clique require all-to-all connections for the subgraph?
 A: They are very different types of object, but there is a relation between them.
In graph theory, an $n$-clique is a graph with $n$ vertices, and an edge joining each vertex to each other vertex.  Note that a graph, in the language of graph theory, consists only of a set of "vertices" and a set of "edges", each edge corresponding to a pair of distinct vertices.
In geometry or algebraic topology, an $n$-simplex is a subset of a Euclidean space which is the convex hull of $n+1$ affinely independent points (called the vertices of the $n$-simplex).
As a polytope, the $n$-simplex has edges which join each vertex to each other vertex.  Thus the vertices and edges of the $n$-simplex correspond to a graph which is an $n+1$-clique.
The use of the same words "vertex" and "edge" in both contexts is not really an accident, as polytopes provided some of the important motivating examples for graph theory.  But in the context of graph theory, the geometry is stripped away and all you have is an abstract set of "vertices" and "edges" that are pairs of vertices.
