Distances and times (and other units of measurement)
Consider a graph where the edges are the driving distances between warehouses across the country. If two warehouses are on the same premises, they could potentially be given a distance of 0. Strictly speaking there might be some distance between them, but this may be considered negligible if the distance is small enough or if you can transport the goods without actually having to drive from one to the other.
Why not just merge those into one vertex? Wouldn't all the outgoing and incoming edges be duplicated? Not necessarily. If different warehouses serve different purposes, it could be reasonable for those 2 warehouses to be connected to different sets of other warehouses.
Or consider some arbitrary process flow graph (e.g. the general flow of processes and equipment inside a factory) with various tasks which each generally take a few days to complete, where the edges represent time.
Now consider a task that can be run in a few milliseconds on a computer. Under the right circumstances, one could reasonably just assign this an edge weight of 0.
As above, you may want to keep the vertices which have a 0-weight edge separate, because this can provide some visibility about the additional process and better reflect the real world, and the vertices can again be connected to different subsets of other vertices.