It's heartbreaking to hear your story. Your older son, I'm sure, is a wonderful boy. Your second son, unfortunately, feels he suffers in comparison and it's making him feel both terrible about himself and furious at his big brother.
It's always hard for second kids. My daughter once said to me "You don't know what it's like growing up with a genius for a big brother." Now, her brother is no smarter than she is. But to her, since he was older, he always knew better and seemed to shine brighter. And when kids are the same gender, the competition is usually even greater.
What second children usually do is carve out new territory for themselves. In other words, if big brother is brilliant, little brother becomes an artist, or an athlete. Often first kids are a bit more reserved and the second child is more of a people person.
It sounds like in your home, your sons share interests. And with homeschooling, your second son may not have as much opportunity to live in a world without his brother. Whatever you think of school, it does give kids a chance to be in their own world without their siblings, and that is usually a very good thing for their individuation.
I also want to address your point that it is people outside your immediate family who are "creating" this problem. Most of the time, a child who feels good about himself will feel good about himself regardless of what he picks up from the grandparents. In fact, he probably won't pick up any negativity from them at all, even if they do slightly prefer his sibling. And if your son is afraid that even new acquaintances like his brother more, then he is going into those interactions with a chip on his shoulder, and is creating those negative interactions. But the bad feelings are also coming from inside him, and the whole process is circular.
Finally, I am concerned about your son's general negativity. "I just hate my face!" and "You hate me!" are indicators that your son is thinking negatively and has some big fears.
So what can you do?
1. Give your son a life of his own. If your boys are homeschooled together, and if you aren't open to changing that arrangement, then be sure that he gets substantial time in venues away from his brother -- in classes or activities of some kind.
2. Start looking for each child's unique strengths and emphasizing them. Every child has some.
3. Facilitate friendships for your children that are separate. Each deserves friends of his own that he can play with without his brother there. It is terrific that your boys generally play well together, but it is not good for his development for your younger son to always be in the position of being the younger playmate.
4. Listen to your son express his upsets. Don't try to talk him out of his feelings. Instead, empathize. "You think Grandma likes your brother better? That must feel so terrible, Sweetie." Notice you aren't agreeing with his opinion, just validating his feelings. If I were in his shoes I would have a lot of crying to do to express how sad and unloved I felt. Help him feel safe enough to cry. Getting past those feelings will liberate him from their grip so he doesn't go into every interaction with a chip on his shoulder.
5. Talk to your relatives, beginning with whoever you feel closest to. Do they realize that your son is picking up that he is less liked? Is there a problem with your son for them -- meaning is it true that they actually have a hard time with him? Do they have any suggestions? Is there a way for him to spend one on one time with the grandparents, for instance, so that he has a chance to bond more deeply with them and feel special? Can the cousin closest to his age and your son spend some one-on-one time together so they bond?
6. Help your son with social skills. First, use the info you got from the relatives. Is there something your son is doing that is making people not like him? Then, think about what you see socially with peers. If he is having a hard time with peers, observe and see what is happening. Maybe he doesn't know how to observe a group and then join it. Or maybe if he isn't as comfortable as his brother with the cousins, he compensates for his insecurity by being bossy. You may need to get some help from a counselor who is experienced in helping kids with social skills, or even find a social skills group for him. What you don't want is for him to be socially insecure as he gets older.
7. Address your son's negativity head-on. For instance, you might "play" with the issue by vying with him to be the most negative possible, as a game. Then, turn it around and compete to be the most positive possible. As long as he's laughing, you're making progress with this.
I also suggest that you get your hands on Tamar Chansky's book Freeing Your Child From Negative Thinking, which is a terrific book to help you help your child.
8. Spend time with him alone every single day. I know you have three kids, but this is something every child deserves, and if a child is having a problem, your loving attention and connection will be a huge help to him. You might want to read about Special Time on my website:
I hope these suggestions are helpful. This is not a situation that will get better by itself, so it's terrific that you're being proactive to help your son. Good luck!