Matching problems that are easy for bipartite graphs but hard for general graphs
The stable marriage problem with ties (a matching problem for bipartite graphs) is (trivially) solvable in polynomial time, whereas the stable roommate problem with ties (a matching problem for general graphs) is known to be NP-hard.  See Eytan Ronn:
NP-Complete Stable Matching Problems. J. Algorithms 11(2): 285-304 (1990) for details.