EdgeConnectivity returns the edge connectivity of a graph, that is the minimum number of edges whose removal disconnects the graph. A set of such edges is called an edge-cut.  You can use the IsCutSet command to test whether a set of edges is an edge-cut.

VertexConnectivity returns the vertex connectivity of a graph, that is the minimum number of vertices whose removal disconnects the graph.

By an elementary theorem of graph theory, the vertex connectivity of a graph is less than or equal to the edge connectivity, which is less than or equal to the minimum degree.