The advantages IMHO are:
1) You can remove the trub from the wort by briefly opening the valve on the bottom of the connical, rather than racking the wort off the trub into a secondary.
2) The whole process can take place in one connical rather than having to move around heavy carboys.
3) There is less chance of contamination, since the whole proccess will take place in one fermenter rather than racking into a second.
4) They look cool, and you can draw off samples without opening them.
5) They are generaly used for bigger batches, hence why #2 is important.