Minimum - cut is easier: you have fixed endpoints and can use max flow algorithms. Global minimum cut is harder because you explore all possible partitions without knowing which vertices should be on which side. Stoer-Wagner is the standard algorithm for global min cut. It is more complex than max flow but runs in polynomial time, making it practical for moderate-sized graphs.
In practice, if you can fix and (like in network flow problems), do so. Global min cut is only needed when the partition endpoints are unknown or you want the overall weakest split.