Welcome to Flyer Talk. I live in Florida - and have stayed at 4 of the 9 Ritz Carltons in the state (Naples, Coconut Grove, Amelia Island and Sarasota). I'm familiar with the other properties.
I think the first issue is - since you're traveling with children - whether a property will be kid-friendly. I don't think Sarasota is. It is more or less a small city (Sarasota is a small city) hotel with a beach club that's about a 10-15 drive away from the hotel - assuming you are in off season and there's no traffic. It's more of an adult place (not only the hotel - but the entire area). Ditto with Naples (although it's directly on the beach). Naples Golf speaks for itself. South Beach is more of a "young adults who want to be in South Beach scene". Coconut Grove is mostly a business hotel.
Palm Beach is a kid-friendly property - especially in the summer - but it is an older property and not to the liking of a lot of people here. Palm Beach is also very expensive. It's a "big bucks crowd". Ditto with Key Biscayne (a newer property - but - again - with a "big bucks crowd"). That leaves Orlando and Amelia Island. Orlando brings the Disney experience into play (which you may or may not want). I happen to live up near Amelia Island. The property is nice - on the beach - recently renovated - with lots of kid stuff - and considerably lower key. On the other hand - we are in north Florida. The ocean - even in the summer - is not hot (it's - at best - warm). Amelia Island isn't the most happening place in the world (although the kids might enjoy the Jacksonville zoo - a pretty good zoo - about a 30 minute drive from the hotel). There are lots of inexpensive restaurants in the area (people in Fernandina Beach have a lot less money than people in Palm Beach or Sarasota or Naples or Miami).
You didn't say when you were planning to go - or what you wanted to do when you got here. So perhaps if you give more information - we can help a bit more. If you are talking about - for example - Christmas - it is too cold to swim in the ocean up north at Amelia Island - although the hotel has an indoor pool and does put together one of the largest gingerbread houses in the world for the kids.
I think at any Ritz Carlton you will have a certain degree of formality (that's just the way Ritz Carltons are) for various things (like for the Christmas tea for kids up here - the kids get all dressed up). You don't necessarily need a huge amount of money - but you're not talking about a Motel 6 budget either. To get the best of the hotel experience - you should check what things cost at the hotel - and plan to include some of them in your trip. And you will have to pack appropriate clothing for various activities. Robyn