This is a commonly litigated - or simply contested - area of law in Florida. However, the case law is clear. You may reasonably trim trees that grow over onto your property line that emanate from a tree that grows from your contiguous neighbor's yards. However, you cut down the branches at your own risk and costs, and you cannot do anything that will cause permanent damage to the tree or harm the tree. For example, you may not even destroy roots that flow into your property if doing so will kill or harm the integrity of the tree. Before you take matters into your own hand, you might consider hiring a lawyer and having the lawyer write a letter to you neighbor and come to a reasonable agreement regarding the tree branching which are encroaching onto your property. This will eliminate, should you come to a reasonable agreement, and litigation and, more importantly, any neighbor dispute, which can be as bad as a divorce.
If you cut the limbs into your area you should dispose of them in your yard. There is no reason to disturb your neighbor or his yard if you are the one that cut them down. Placing them in his yard for disposal can lead to liability and litigation
I agree with the previous answers and if you cut them off then you should compost them. I always recommend hiring an arborist who is licensed and bonded to deal with vegetation at the property line because their insurance will cover you if there is an issue later.
Good Luck
You can pick fruit off the branches of your neighbor's tree that hang over onto your property or you can cut off those branches because they are considered your property. You cannot dispose of your property (including the trees you cut off your neighbor's tree) by throwing it onto your neighbor's property. If you do so, you can be liable to your neighbor for trespass. If you are unsure of your rights and responsibilities, you should consult an experienced real estate lawyer in your area.