There should be basically no differences between (the latest version of) Server and Cloud. For older versions of JIRA, check the differences between the /latest documentation and your particular version:
https://docs.atlassian.com/jira/REST/
OK. So that means after we do some work to integrate a local APP with JIRA Server version(mostly getting data from JIRA through REST API), it should automatically work with JIRA cloud version if we choose latest version of JIRA, right?
I disagree. I have just hit a difference - you cannot expand properties on an issue search with server rest, but you can in cloud rest ... why ?
Please note that in the last few years this has changed, due to Cloud moving to a multi-tenant architecture. The Cloud REST API and Server REST API have diverged somewhat and are now different. Please compare the following pages:
https://docs.atlassian.com/software/jira/docs/api/REST/latest/
https://developer.atlassian.com/cloud/jira/platform/rest/v3/
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