In Florida, when a healthy tree falls in a storm the damaged property owner normally claims on their own insurance, regardless of whose land the tree grew on. Responsibility shifts to the tree's owner when the tree was visibly hazardous and they were on notice and did nothing. This is general information, not legal advice.
The healthy-tree rule
A healthy tree brought down by a storm is generally treated as an act of nature. Your insurer covers your damage; your neighbour's covers theirs. It feels unfair when the trunk plainly grew on the other side of the fence, but that is the usual starting point, and it is why arguing about it rarely gets anyone anywhere.
When negligence changes it
The picture changes if the tree was obviously dead or defective beforehand and the owner knew or should have known. Written notice is what turns 'should have known' into 'knew'. If a neighbour's tree genuinely worries you, a dated letter with photographs is worth far more than a conversation over the fence, both for prompting action and for the record afterwards.
Branches and roots crossing the line
Florida generally lets you trim branches and roots that cross onto your property, at your own expense, back to the property line. What you may not do is cross the line to do it, or cut so aggressively that you kill the tree, which can make you liable instead. If in doubt, have it done properly and keep the invoice.
The practical advice
Most of these situations are cheaper to solve than to litigate. If a neighbour's laurel oak has large deadwood hanging over your roof, an assessment costs nothing and often gets the tree dealt with once both parties can see it in writing. We are happy to look at a tree and say plainly whether it is a hazard, and that assessment is the same whichever side of the fence it is on.