What is the difference between graph-partitioning and graph-clustering in graph theory?
Graph partitioning and graph clustering are informal concepts, which (usually) mean partitioning the vertex set under some constraints (for example, the number of parts) such that some objective function is maximized (or minimized). We usually have some specific constraints and objective function in mind. However graph partitioning and graph clustering, as vague informal concepts, are pretty much the same.
There tends to be an emphasis on edges in partitioning. ("A good partition is defined as one in which the number of edges running between separated components is small." from the English Wikipedia.) On the other hand, clustering tends to be about vertices (or the connectedness of the subgraph of neighbors of a vertex). This is entirely a linguistic artifact and I would not expect it to be as prevalent in non-English writing.
(Of course, it's challenging to separate the roles of vertices and edges in defining nearness or connectedness, so such a distinction is necessarily somewhat artificial.)
Both creates partitions/blocks. In partitioning number of partitions/blocks 'K' is given in advance. While in clustering we do not know what number of partitions/clusters/groups will be formed by clustering. Following screenshot is taken from this text.