For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular, the unit ball is compact in both topologies.
One problem with the weak operator topology is that the dual of B(H) with the weak operator topology is "too small".
The ultraweak topology fixes this problem: the dual is the full predual B*(H) of all trace class operators.
In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient.
If H1 is a separable infinite dimensional Hilbert space then B(H) can be embedded in B(H⊗H1) by tensoring with the identity map on H1.