I run Windows 7 64-bit. I have the Java JRE 32-bit installed. I use Firefox almost exclusively for web browsing. I see there is a 64-bit JRE, but I can't really tell what use it is. Under what ...

Back in the early 1.5 days, yeah, the JVMs were buggy for us. Since we've been using late 1.6 VMs from Sun -- it's been fine. No issues whatsoever.