Fedora Core is more cutting edge, where red hat/centos is more geared toward being a stable enterprise OS. If you are wanting a desktop environment I would say Fedora Core for all the latest software and updates. For example they release a new version every 13 months, and X+2 updates every 6 months.

Get Red hat if you want to pay the licensing fees, or Centos if you can wing it yourself.

However I would go with centos if you are wanting a stable server that is RPM based. Also a good distro I find for servers is Ubuntu server; setting up and upgrading is a breeze, plus Ubuntu has tons of community support.

I guess it all comes down to what you are looking to do. That would better help determine what the advantages and disadvantages are.