What are the advantages of having USB part of the microcontroller (as the Leonardo does)?
Speed!  You can send out up to 1000 frames of data per second of data to the 32u4 using it's virtual serial port.  The older boards with their FTDI chip and UART bridge are limited to around 38 kbaud before signal reliability problems start to set in.

You can also represent more devices than just 'com port' -- for instance emulating a keyboard and a mouse.  If you get fancy and use a new firmware like LUFA you can appear to be many more types of devices.