In ring theory one often wants to think about bimmodules as being morphisms between rings using tensor product as composition. However, this composition is only associative if one uses isomphism classes of modules.
In other words, the category BiMod is defined to have rings as objects and isomorphism classes of bimodules as morphisms, and if ${}_R X_S$ and ${}_S Y_T$ are bimodules, we define the composition
$$ [ Y ] \circ [ X ] := [ X \otimes_S Y ]$$
where $[ \cdot ]$ denotes bimodule isomorphism class.
Another approach is to use the concept of a "bicategory", which is exactly designed to deal with situations where we have a notion of "isomorphim" for the proposed morphisms in the category and composition is only associative "up to isomorphism".
Bicategories satisfy certain axioms, whose ultimate consequences I am still working to digest.
My question is: What is gained by taking the bicategory perspective instead of simply using a category whose morphisms are "isomorphism classes" of bimodules? Do the bicategory axioms imply something more than what is obtained simply by using a category whose objects are isomorphism classes" of bimodules? Or are the two notion essentially equivalent and the bicategory approach is simply provides a "perspective" that some people find more enlightening?
I'm a novice to bicategories, so any clear explanations (even at the risk of being pedantic) are much appreciated.
In general, the 2-morphisms of a bicategory don't have to be isomorphisms. They can be e.g. inclusions, as in the case of Rel, the bicategory of sets, relations, and subrelations.
This is something of a facile answer, but hopefully it will suffice as food for thought while somebody writes a better answer.