Thank you for using JustAnswer.
I'm sorry to hear about your situation.
The health of the tree determines which landowner is responsible for damages to property (1 Fla. Jur 2d Adjoining Landowners section 8 ). In the situation where a dead tree falls on an adjoining property and damages that property owner's home, the landowner who owns the property where the tree originally was located is responsible for damages. Alternatively, in the situation where a live tree falls on an adjoining property and damages that property owner's home, the adjoining property landowner is responsible for damages. Put another way, consider Landowner A (property owner of tree) and Landowner B (adjoining landowner). If Landowner A's dead tree falls on Landowner B's property, Landowner A is responsible for damages. Conversely, if Landowner A's live tree falls on Landowner B's property, Landowner B is responsible for damages.
In short, the fact that it was dead made it the responsibility of the landowner to responsibly remove the tree so that it would not harm you or your property. Failure to do so would make your damages your neighbor's responsibility.
When you actually fix your screen and replace (or have an estimate of the fair market value of the equipment), you might be able to take that to small claims court if the amount is under $5,000. If more, you would need to get an attorney to pursue that case, as it would be in a more formal court of general jurisdiction.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Thank you, ***** ***** luck to you!