How is testing an application, that is hosted on cloud, different?

It highly depends on level of testing you are performing.

If it's just functional web testing (Black box) - most probably you won't notice any difference.

But if you are performing some more technical testing - like some API testing - you will need some knowledge of cloud infrastructure your application is using. It have nothing to do with your application functionality, but you will need to get logs from cloud services and so on.

Also you will need knowledge of cloud infrastructure if you are involved in deploying builds or involved in Grey/White box functional testing.

Some applications also use specific cloud infrastructure provider features like custom data storage, queues and so on. These custom features usually doesn't bring anything new to testing, but sometimes your application features can be highly tied to infrastructure features - and you will need to understand how it works.

Testing a cloud application remains same as you would do for an on-premise data center hosted application.
However for an application hosted on cloud the main consideration you need to take care of is the network connectivity between you testing workstation and the cloud application endpoint.

Apart from this if you are trying to do security related penetration/vulnerability testing for your application then you need to  inform the cloud provider that you are planning to carry out such an activity on your application and need to fill forms and get permission for executing the same as the virtual servers hosting your application are on a shared hardware and so require prior intimation.