Installed CWM on note 2, but the phone is not rooted
You are rooted. AFAIK Samsung bootloaders are already unlocked, you only need to flash the recovery to get the "root"

Now for the problem.

Go to settings-&gt;about phone and tap the build number 7 times to enable developer options. Then go to settings-&gt;developer options and there you should be an option similar to root access. Make sure you select Apps or Apps and ADB (recommended) in it and you should have root access.

If there is no such option, download and flash SuperSU via recovery and you should have root access later.

And that addition of "255|" in indicates you being in root user mode and accessing terminal as root. So its a positive sign here as well.

Alternatively, in case you failed to get the root to work, you can also flash 
some custom ROM like Cyanogenmod via recovery.

But rest assured, you are rooted. Its probably some bug and flashing supersu should probably fix it.