The Wireless Toolkit is a way to see how your J2ME application is working. it uses an Emulator to emulate a "real" J2ME device. In theory, what runs in the Toolkit should run on any J2ME compliant device. In reality though, some manufacturer's don't quite implement 100% to the spec, which makes the Toolkit a good way to do quick testing, but when it comes down to a real device, it is test there too.
Each phone needs an operating system to run, regardless of J2ME or not. Verizon Wireless phones strictly run the BREW OS. Whereas most others use the Symbian OS. It is kind of like the difference between Windows and UNIX. So with that in mind the J2ME "Sandbox" runs just like the J2SE JRE on Windows or UNIX. It is platform independent.
So the relation between Sybian and BREW is that they are Operating Systems. J2ME can run on top of both OSs. But Symbian I think is more friendlier and most of those devices already have the JRE installed on them. On most BREW phones, you have to download the JRE to run J2ME MIDlets.