If you're comparing Emergency Dumpster Rental to a standard dumpster rental, start here. It is a container needed at short notice because something went wrong. You'll want this settled before the truck shows up: water-damaged material is dense, so the container is sized on weight rather than the visible pile.
A box of old carpet and a box of broken tile can be the same size and still hit completely different limits. One runs out of room, the other runs out of allowance. When anyone dealing with damage call in this kind of job, the debris volume is usually easy to anticipate in advance, so the containers that get dispatched tend to land in the 20 or 30 yard bracket. What decides the call quickly: the width of the path a truck needs, and whether the container sits on sidewalk that needs protecting.
With the scope not yet fixed, what ends up in the container is hard to know ahead of time, so treat the first size as a starting estimate rather than a final order. Find out if a locking lid or gate is available for a container that will sit on the property overnight in a neighborhood with foot traffic. The dumpster you actually need is the one sized to the heaviest material in the load, not the one that looks roomy enough for everything else.