Cache is often the culprit in things like that. Followed by plugins.
It is also possible the added code that broke some browsers. You could always open a ticket, especially if you can recreate the issue on a clean install with no plugins.
I've sometimes had success with switching to private mode when things were acting weird, but haven't had to do that often.
Pretty sure that they added new code that doesn't run perfectly well in older browsers. I'd send a ticket but I don't think they'd care since that might go against their interests. It seems that they'll just force everyone to use current browsers.
I only use Firefox in private mode with all caching functionalities disabled, so cache is hardly an issue for me. Might be for others, though.
The Gecko version in Seamonkey 2.40 is a much older version than the Gecko version used in Firefox 45. Gecko versions are kept in sync with Firefox versions, so the browser you're using is actually equivalent to Firefox 7.
https://developer.mozilla.org/en-US/docs/Mozilla/Gecko/Versions
EDIT: Sorry, my bad. Didn't discern 2.40 from 2.4. If the order in this link is maintained for all the following releases, Seamonkey 2.40 uses the same Gecko engine as Firefox 43, so much more recent. Might still not be current enough to run the website smoothly, though. I was having issues with Firefox 43 myself.