The main advantage would be the ability to easily share the catalog and photos with a second computer by just moving the external drive.
As a general rule, with exceptions, internal drives tend to be faster than the external drives due to the USB connection of the external drive.
And the upside to having the catalog and all images and associated LR files (presets etc) on a dedicated external drive is cloning it to other dedicated drives. So maybe a bit slower, but much easier to keep all this important data in one place for both cloning and as ManiacJoe says, sharing. With a dedicated drive for images and LR files, you can hook it up to any machine that can run the version of Lightroom you're using.
Obviously, backing up the files is a must.
However, what does cloning the drive do for you?