Can neighbor sue me for damages after insurance paid?

If you're talking about his or her insurance company, then probably not since it's customary for the insurance company to get subrogation rights in exchange for payment. The insurance company can certainly sue you but they don't always do so.

OTOH, if your insurance made the payment then absolutely not unless your insurance company is incompetent. Insurers always make claimants waive all future legal remedies arising from that claim before paying. The claim is considered settled in full.