Hello. My name is Marc. I'm a licensed attorney and I will be happy to answer your question.
I don’t know what the basis was for your insurance company to deny your claim. Perhaps, as is often the case, the insurance adjuster doesn’t know the law. Or perhaps there’s an applicable exclusion in your policy. In any case, the adjuster was wrong to tell you that your neighbor is liable for damages.
In SC, for a tree that is on urban or residential property, then your neighbor has a legal obligation to prevent risks from unsound trees. Thus, if a branch or tree falls and harms a person or property, you still have to prove negligence.
You can prove negligence by showing your neighbor knew or should have known the tree created an unreasonable risk, and that risk caused the injury or damage. For example, if your neighbor’s tree had noticeable disease or decay, was uprooting, or frequently lost limbs, then your neighbor probably knew or should have known there were problems with the tree.
If the tree is perfectly healthy, but a lightning strike causes the branch to fall, then the neighbor probably isn’t liable for any property damage or injury under the “Act of God” defense.
So, in your case, I understand that the tree was live. Do you have any reason to believe it was unhealthy or otherwise compromised? If you can establish anything like that, then your neighbor should pay. I'm going to assume, however, that the tree was fine.
Did this occur as a result of Hurricane Matthew? If so, then, whether the tree was healthy or not, the Act of God defense may apply and your neighbor is not liable.
Finally, perhaps the insurance carrier had another reason for denying your claim. Policies often have exclusions for acts of god, for example, which might apply here if the tree fell because of the hurricane.
In sum, consider calling your insurance carrier back. From what I can tell, there is no basis to deny your claim because, under SC law and the facts as I understand them, your neighbor is under no obligation to pay for damage caused to your property by his tree.
I hope I have provided you with useful information sufficient to help you better understand your issues and options. If so, please feel free to rate my answer, as that is the only way I can receive credit. Meanwhile, good luck dealing with your insurance carrier.