As far as I can tell, the max cut and min cut problems seem to have different inputs: max cut enumerates over all partitions of the vertex set into two subsets, while min cut has a specified source and sink.
Also, if
If you try to take the complement of the edge set to translate a max cut problem, you end up not minimizing cuts, but cuts plus minus a constant correction factor that depends on the size of the subsets in the partition (namely, it is the product of the sizes). This changes the nature of the problem considerably - if you have a dense graph, min cut will tend to separate a small chunk, while the corrected version will not.
Edit: A previous version complained about a difference in input type, hence David's comment.
