Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)?

A $*$-homomorphism between two von Neumann algebras is weak* to weak* continuous if and only if it is ultrastrong to ultrastrong continuous. See Proposition III.2.2.2 of Blackadar's book (which, basically, answers all questions of this type that you might have).

A von Neumann algebra is a $C^*$-algebra $A$ that admits a predual, i.e., a Banach space $A_*$ such that there is an isomorphism $A\to(A_*)^*$.

A morphism of von Neumann algebras is a morphism of $C^*$-algebras $A\to B$ that admits a predual, i.e., a morphism of Banach spaces $B_*\to A_*$ such that $(A_*)^*\to (B_*)^*$ is isomorphic to $A\to B$.

The weak topology induced by the predual on $A$ is precisely the ultraweak topology, and so ultraweakly continuous morphisms are precisely those morphisms that admit a predual.