Matt, best solution for you will be to review Atlassian's release notes. Each organization has different needs so the benefits will change accordingly.
Go to https://confluence.atlassian.com/display/JIRA/Production+Releases and work your way up or down to understand what new features were added (quite a lot in your case), what bugs were fixed.
I can point out that you'll have the ability to use a lot of updated plugin versions, which is quite important to most users.
More functions, improvements to existing ones, and lots of bug fixes and enhancements.
I'd read the release notes for 5.1, 5.2, 6.0, 6.1, 6.2 and 6.3 - the main landing pages for them will give you the highlights of each release
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