It's not either/or--it's both. That is, if you have relevant insurance coverage, you could certainly file a claim. If you do not, or choose to not file against your own insurance, you could ask the neigbor to pay; if he or she does not, you could sue him. The lawsuit would be based in negligence--that your neighbor's unreasonable carelessness in how/where/etc. he shot the gun damaged your property. You could sue (or claim under your insurance) for the actual cost to repair the damage done by the errant shot.