You can see the version history for JIRA Software at https://confluence.atlassian.com/jirasoftware/jira-software-7-0-x-release-notes-776997721.html and for JIRA 6.X at https://confluence.atlassian.com/jira/jira-release-summary-159154300.html.
These also describe what was different between the versions - a clear note of warning for version 7.0.8 on mySQL but apart from that all versions should be stable.
"Stability" is pretty much irrelevant in this choice, you really should be making the choice based on requirements (although, I tend to avoid the very first release of a major version - 7.0.1) I'd always recommend 7.0.latest over 6.x though, as it's got better functions, and you'll end up upgrading the 6.4 to a later version eventually, might as well do it now.
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