Pros
You don't have to pay for your tickets until final payment for your cruise. If you cancel the cruise before making final payment, you owe nothing on the air.
Some people feel a sense of security thinking that FBC has an obligation to make sure you get to the ship if things go wrong (e.g. your flight just got cancelled). I am not among those people. Here is a positive testimony: link.
Another potential pro is that sometimes FBC airfare is cheaper than if you booked the same flight yourself independently. This used to be very true when it came to booking one way long haul flights for a transatlantic cruise for example. Other times the airfare is cheaper when Celebrity is running a promotion tied to booking a cruise ("$200 off airfare", etc).
Cons
You have to work through Flights By Celebrity if you need to resolve a problem with airline (e.g. your flight times have changed and now you have a connection that is too tight and you want to change to a different set of flights). I have read stories where Flights By Celebrity provided great service in this area (e.g. FBC can book you on a different airline altogether if there is something wrong with your flights) and I have read many stories where they were a hinderance on something that could easily be fixed by you talking directly to the airline. Here is a very recent and heavily documented example: link
Phone wait times with FBC can be very long (I waited an hour on a few occasions) although that can also be said about phone wait times when you call the airline.
Personally for me, I don't like the idea of having to work thru FBC when I could otherwise call the airline myself (or do it on the airline website) to resolve a problem. However if getting air fare thru FBC represented a large savings or if I am hesitant to buy airfare for a cruise I am not 100% sure about, I would consider going the FBC route.
Seats
As far as choosing your seats are concerned, it depends on the airline. I know when I booked my United and Delta flights through FBC, I was able to choose my seats. I have heard airlines like Air Canada do not allow it. Here is some more information on the subject: link