Hm honestly when it comes to threats, it's time to do something. A normal person would come up to you and talk to you in a calm voice, discuss what they find a problem and allow you to defend yourself or make it right, apologize and assure there will be no future problems, etc.. What he did was just have a tantrum first and because you didn't back down to it [as I'm sure he's use to getting many to do], is now focused on you and obviously it's become a obsession.
I once asked a neighbor, after two years of taring up my grass, to stop parking on my lawn. I was extremely polite and said my pleases and thank yous, I couldn't of kissed his back end any harder. But it didn't matter, they didn't park on my lawn anymore but they also made it clear how mature they were...they started tossing trash in my yard, making extremely loud noises all times of night, threw things against my house and so forth. I just picked up the trash and pretended it didn't bother me. If I could do it again, I don't think I should of just sat back and took it. I don't think you should either, especially when he is threatening with physical harm. Never underestimate a person, especially one being so aggressive. He could find ways to take it out on your dog, even set poison out, etc.. People with this childish mindset don't use maturity or common sense. They do drastic and unreasonable things.
Start recording his threats and call the police. Make your landlord aware, etc.. No one should have to put up with that, plain and simple.