While both are Southern Coastal cities, they are very different. We enjoyed them both. 
We stayed in Charleston for months and visited what we could on weekends. The downtown was strictly shops and stand in line for restaurants. Their City Market is outstanding and it's really a walkable city. A free trolley bus took us from the Aquarium to downtown and back. 
We've been to Savannah at least 3X. A couple of day trips and one extended stay when we took my Mother after my father passed. The trolley took us around the first two trips. That way we got orientated and educated on which movies were filmed where and how the city was laid out. With Mom, we used it to get on and off at historic home tours and restaurants or shopping. Then back to parking lot and the motel.  
The 3rd time, we went to Fort Pulaski and the Old Fort Jackson. We even saw a cannon shooting demonstration. Very interesting and informative. Both were on the way to Tybee Island for seafood lunch and gator viewing.