Under the common law of torts, your duty is to keep the tree and its roots off of your neighbors land, and to use due care to not impose an unreasonable risk of harm to your neighbor's property.
For a trespass action, your neighbor will be required to show that the tree roots are actually physically entering his land, and that the trespass is the cause of damage to his driveway and garage.
Here, it will be easy to show that the root has entered his land. And, to the extent that the root causes damage, such as causing the ground to be unlevel, then you would be strictly liable for the damage.
However, for you to be liable for the damage to the paving and garage pad, your neighbor will have to show that you were negligent: that you owed a duty of due care to the neighbor, that you breached your duty, that the breach actually caused the damage to your neighbor's property, and that the neighbor was damaged as a consequence.
Here, your neighbor must show, among other things, that the root is a "substantial factor" in causing the damage to his driveway and garage pad. This may require him to hire an expert to testify as to the cause of damage, which will cost him at least $2,000 in expert witness fees that he cannot recover from you at judgment. When he speaks with an attorney about his case, he may not want to pour money into a risky legal action. Furthermore, your neighbor cannot collect attorney fees from you, so unless an attorney thinks that this case will be easily won, the attorney may not be interested -- which gets your neighbor back to fronting the money for an expert to report on the cause of damage.
I don't want you to think that you're "out of the woods" here (pun intended). An attorney would probably be willing to send a nasty note and your neighbor may be willing to file a claim just to get you to have to retain a lawyer to defend, which means that you will also have attorney fees you cannot recover. At which point you will have to choose between settlement and attorney fees -- neither of which will be free.
I think you may want to offer to remove the root from his yard, in exchange for his releasing you from further liability concerning any existing tree damage. He will probably refuse, but one never knows.