You are losing the battle because you are losing the connection. Your son is not acting out because he wants to inflict pain on his brother, but because of a deeper reason. It's your job to figure out what is causing the biting and aggressive behavior! Punishing him only makes it worse!
I know it's not what you wanted to hear and not what most people will tell you! He is expressing a need (in an inappropriate way, yes) and he may be frustrated, angry, mad, whatever, but he is feeling like you are not understanding him at that moment. Maybe he doesn't communicate well? Not speaking much? And has strong emotions! You need to try to connect with the reason and let him know you get it before he will stop acting out in that way.
For example, you see the 2 boys struggling over a toy and one bites the other. You rush over and grab the biter and put him in your lap. You may need to comfort the other one as well, make sure he is OK. Then talk to the biter. "I bet you wanted that toy. I know, it's a really great toy! But your brother was playing with it and now he's sad because you bit him. We don't bite (hit, kick, push, etc) other people. Next time you want what he has, you could try offering him a trade. Give him the fire truck and ask for the airplane. I'll help you."
I don't know how much of that will get through to the child, but go for it. They seem to understand that you are trying to connect and see things from their perspective! That's all they are looking for! Even as they get older and can express themselves with words, it's still hard. Maybe he really wants to climb the ladder and go down the slide by himself but is struggling, then another kid climbs past him and sits down to slide and he bites or hits that kid! You need to say it's not OK to hit, but also acknowledge that he was having a hard time with the ladder and might have been frustrated when a bigger kid passed him. Just relate to his struggle and that will get you far!
Oh I could recommend some great books if you are a reader! Let me know if this sounds like something that resonates with your values and I will make a list. :)
It's too bad you got such poor advice like biting him back or putting vinegar in the child's mouth! Yelling, hitting, and time outs are also inappropriate. Kids learn by example, so start setting a good one in how you handle things. If you want a child to learn to hit when he has a problem, then go ahead and hit him! If you want him to fear you or resent you (and not learn that what he did was wrong) then yell at him or put him in a time out. But don't complain when the problems continue because you didn't teach him any coping skills! All parents should connect with your kids, get on their level and try to see things through their eyes! There are many studies that illustrate how punishments do not work!