I am using Firefox 27 now on my computer with 890 MB of RAM, when for whatever reason I cannot use Opera. I haven't noticed that it would be slower.
Web sites themselves, on the other hand, are [much] more bloated, and consuming memory. Look at this one, since the upgrade of the forum engine, and compare it to say Hydrogen Audio. Those sites also "need" the latest browser for basic things like text input to work, whether we like it or not.
I've disabled a few things in my Firefox, such as 'safebrowsing' and the autoupdater (for negligible impact), and moved the cache.disk.parent_directory to a temporary partition to decrease the effect of fragmentation. I'm also using an older version of the Flash plugin, put into the program directory (again, no autoupdater for the Flash). If you can accept that some sites will be disabled, try running the browser without Flash at all.