The two graphs of the title both have vertex set G. In the intersection power graph, x and y are joined if some non-identity element is a power of both; in the power graph, x and y joined if one is a power of the other. Thus the power graph is a spanning subgraph of the intersection power graph, and we define the edges of the difference graph to be the difference of these edge sets. In this paper, we give a number of results about the difference graph. We examine groups whose power graph and intersection power graph coincide. In addition, we make some observations on isolated vertices in difference graphs. We study the connectedness and perfectness of difference graph with respect to various properties of the underlying group G. Furthermore, we investigate the operation of twin reduction on graphs, a technique that yields smaller graphs which may be easier to analyze.