I've seen many different funding schemes due to external grants/fellowships. Here are some possibilities, but it seems to depend on the school! I put the department name in case it might vary by that too! Maybe this thread could be useful info for prospective students!
Caltech (GPS): I was told that every student gets paid $29k/year, no matter what fellowships you bring in. The only exception is if you have a *private* fellowship that is paid directly to you instead of through the school, then you keep all of the extra money since Caltech is not involved. I don't know what happens if your fellowship is >$29k (e.g. NSF), but it might be like Cornell, see below. My admission letter offered a fellowship for the first year, but if I were to get an external one, I can defer that fellowship to a later year, thus extending my years of guaranteed funding by one.
Cornell (Astro & Space Sciences): Every student has an upper limit on funding. If you have more funding than this limit, they actually REDUCE the amount of your tuition award and apply your NSF towards tuition instead, so you don't see any extra money. Boo!
U. Arizona (LPL): They don't make concrete statements except that "every student with a fellowship will have a stipend equal to or greater than what they would have gotten without a fellowship".
Canadian schools: There is usually a multi-tiered funding level, where you get different stipend amounts based on your fellowship status. You usually get the same or more money while TAing less. See an example: http://www.phas.ubc.ca/graduate-program ... al-support
And as said above, the advantage is not having to worry about your funding, and not having to spend time working as a TA! Your TA work load will probably be reduced or eliminated, so that instead of earning your stipend through TA-ing, your fellowship "pays you" to do research/course work.