Difference with respect to JS files inclusion(based on the screenshot shared in the query) are the ones from /libs/clientlibs/granite/coralui2/optional folder and modern.js (included in server and not in local)
modern.js from /etc/clientlibs/granite/loadash/modern.js is not available in 6.5.0
Also, handlebar.js from .../optional/clientlibs/handlebars.js is not available in 6.5.0 (clientlibs folder within optional is not there at first place)
High level root cause for this would be use of coral ui 2 resource with respect to RTE perhaps say custom plugins, leading to inclusion of /libs/clientlibs/granite/coralui2/optional/rte that has dependency on handlebars.
Given all this, I suggest to check the following
Availability of files in these locations in both server and local.
Latest project specific code base on both the instances.
If you have overlayed clientlibs, cross check its availability in both the instances (as I couldn't see above mentioned in my plain 6.5.0 instance)
Comparison on server and local on exact same page (Server is from live copy page and local is from lang master from the screenshot).
You have already compared the instance details and confirmed it to be 6.5. Just to be sure, cross check from http://localhost:4502/aem/start.html -> Right hand section -> "Help" -> About Experience Manager. (If you have checked it this way already, ignore)