In cloud hosting, the biggest difference is how resources are allocated and how many servers you have.
With shared hosting, you get a single server, and all sites on it share its resources, but since there is no limit on how much CPU and RAM each site can use, any site may do so to its own advantage at the expense of others.
In the cloud, your site gets hosted on a pool of servers, which improves the redundancy, and their shared resources are virtually partitioned so that you get a dedicated amount that no one else can use.