What can be solved with semidefinite programming that can't be solved with linear programming?
A typical problem is MaxCut: output a cut in a graph that (approximately) maximizes the number of edges cut. Goemans and Williamson showed an SDP approximates the value of MaxCut to within a factor at least 0.878. Recently, Chan, Lee, Raghavendra, and Steurer showed that for a natural linear encoding of the MaxCut problem, all polynomial size LPs achieve approximation no better than 0.5. 

It's hard to say concisely what kind of problems usually benefit from an SDP. A systematic approach to constructing SDP relaxations is through hierarchies, the most powerful of which is the Lasserre hierarchy: see Rothvoß's survey for a nice introduction. By now there are too many examples of successes of SDPs in optimization to list. Also, Raghavendra showed that one particular SDP gives the best approximation to all MaxCSP problems, if the Unique Games conjecture is true. 

Check the books of Gaertner and Matousek, chapters 6 and 13 of Willimson and Shmoys' book, Lovasz's survey.