Do Caribbean islands typically get worse destruction from hurricanes than bigger places such as Florida?

Yes, but not because Florida is physically bigger.

As a hurricane-prone area, Florida has very strict building codes for all new construction.  Buildings, commercial or residential, have to be designed and built to withstand expected weather events.

Many Caribbean countries, unfortunately, do not.  Many do, but many do not.

Think of the story of the little pigs and the big bad wolf.  The wolf is any hurricane, and which buildings are more likely to survive?  Straw, sticks, or bricks?

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Some are big with mountains. Some are flat and small.

Not sure how you define 'worse'. Are you going by dollar value of what was destroyed? Days without power, water, or other utilities? How long until everyone has a place to live again and all businesses are open? How many people died? Caribbean nations, on the whole, are poorer than Florida. That means that a destroyed house in, say, Abaco, may not be worth as much as a house in Florida, but it's also true that there may not be as much money easily available to build another one..

The damage from a hurricane depends upon many factors. The size (miles across), intensity (Cat #), and movement speed make major differences. Likewise, the terrain of the land hit (flat, hills, mountains, etc) make a big difference. A flat island, such as the islands in the Bahamas or the Caymans will get more storm surge flooding than a mountainous island like Jamaica. However, the mountains of Jamaica will force much more rainwater from the storm and so get flooding from the water running downhill.