Well I have lived in BOTH places. I lived in Sarasota in grade school- and my parents are buying a place on Siesta Key as we speak, so we still visit there a ton.
The pros to Sarasota are the beaches. The sand is white and powdery, the water is warmer (on the gulf), and it is more relaxing.
Ft. Lauderdale is a much bigger city- the beach is nice still, but because of beach erosion is no where as big (sand leading out to the ocean may only be 30 feet or so in places) and the sand is darker with more 'stuff' in it. The shopping in Ft Lauderdale is much better, and the life is faster paced.
If you are looking to relax- go to Sarasota. Many mentioned great places to go- Siesta key is amazing, ringling brothers house, Mote Marine, St Armand's... just nice places to walk around.
Ft. Lauderdale has a great mall right by the beach- Galleria. It has Neimans (handbag and shoe section under construction right now though), Saks, etc. Always something going on- loud music by all the little spots on the beach etc.
Hope this helps!! Ask if you need anything else!