That makes three. I would not bring a man in my life that caused conflict over my child. If your child is indeed your world, then this is not the right match for you. I'm being slightly blunt and not very flowery in that but these issues get worse. You are your boy's mother and know in your heart what is best. Now your brain has to agree so that you can ask this man to move out and make a more peaceful home again for your son. good luck
I agree with AH you need to put your son first all this fighting is not good in his life especially if it is about him. I doubt he will change and his son will always be favored .Good Luck
I think it may be time to seriously reconsider your relationship with your boyfriend. The reason being is that in the last six months, all of you have had some very drastic life changing events that are very stressful. He has moved in with you and brought his son along (how old is his son, by the way, and does he have full custody of his son or does the boy only stay with you on certain days?) AND he's lost his job. You're already stressed with a full time job and trying to be a full time parent and balance a relationship with a man who is lashing out for whatever reason at your son.
And back to the issue of his son getting away with the same stuff your son doesn't by him, may be due to the fact that he doesn't get to see his son each day if the boy lives with his mother and he only has visitation, so he feels he needs to be more lenient--and same from you with your son because you don't feel you get enough time with your son as you work full time.
So I think this all ends up being a failure to get on the same page with each other because no one gets the time they want or need with each other to really understand and work together. You both are running on stress and a hint of guilt towards your children in feeling like you're not there for them enough.
Probably the only way to diffuse the situation is to either separate living quarters again so you both have your own space to do as you like, and raise your kids as you like. Or just break up because, if the situation with your boyfriend and his son is that of him having full custody and his son lives with him full time, then obviously he is favoring his child over yours, and that is unlikely to change, which will only create further conflict between you all as time goes on.