With self-hosted FusionAuth, you are downloading the software, starting up a server (VM in the cloud, in your data center, wherever) and running it yourself. You are responsible for monitoring, backups and everything else to keep your FusionAuth instance available to your customers.
With all the cloud options, we run FusionAuth for you. Each level of cloud has a different set of constraints, as outlined in the pricing links (click on the ? for more details), but in general we take care of operating FusionAuth for you. You still control the upgrade cycle, but much of the other maintenance is handled by us.