Yes, they do. Data mining is the extraction of implicit, previously unknown and potentially useful information from data (Witten and Frank). You can see a complex network as a dataset, which you can analyse for discovering clusters, patterns and other useful information, such as the importance of nodes according to a variety of network centrality measures.
In particular, you can visualise and visually explore a large and complex network. The branch of information visualisation that deals with it is known as graph drawing and there is an annual symposium in Graph Drawing and Network Visualisation (GD).
Yeah, Both are different. Data mining is simply retrieving useful information whereas complex network is basically describes the structure of clusters of nodes or something. So both differs in terms of techniques and approaches but there may be some common algorithms for both of them.