Despite the name CLOUD technologies behind are rock solid.
CLOUD solutions can lower your ownership cost and increase the availability of your website in general.
Storage - all big CLOUD providers offer redundant object storage. Translates into the fact that your files are protected from being lost. Used with encryption can give you peace of mind that you have fewer chances to lose your precious files than using your own computers.
Database scalability - CLOUD providers may offer scalable SQL's clusters services that probably is better than yours.
API's - a lot of depending on CLOUD provider. Using API's you can create very fast modern applications that connects to CLOUD resources for different needs.
For anybody concern, when you talk about hosting your website into the CLOUD that means your files are stored as object files and served by many web servers at once. If the solution does not comply with these two requests, the solution is standard infrastructure even if it is offered by a CLOUD provider.
Your website CLOUD hosting should provide next in order to give you a better solution.
- load balanced access to your website (many web servers serving your pages at once)
- object storage for your website files (HTML, js, css, any)