Pre-warning - I have little experience as a DM dealing with difficult players, but I'm getting pretty used to dealing with difficult tables in general.
A couple of things we found really handy for our horde of uncooperative players:
Talk it out / Remind him
You said in the comments that:
I have talked to him about it before. He seems understanding until a situation occurs where it basically sets it off
So he may just need a gentle, at table reminder about the discussion you've had. A simple prod in the right direction might be all he needs to cool his head. DnD can get pretty intense at times, and it could just simply slip his mind as he gets into it.
Players really getting into your campaign is what you want as a DM, as long as they're getting into in good spirit.
Is the table okay with this behavior?
Is it just you that's bothered by this? How your table reacts to his behavior could also dictate how you handle it.
During our romps with difficult players, one player refused to cooperate with the DM to the point we almost had to kick him out. But the DM pulled me aside, and asked me to respond to every that player said as in character.
If your brother is heavily into role playing, responding in-kind and keeping it to your characters may help stem the OOC poor behavior.
I.E - If he's complaining about the Gods not lending a hand, a NPC (or other player) could take that at face value. "[Gods Name Here] wouldn't care about a single [town/city/country/]! Our lifes are nothing compared to the grand scheme of things. The immortal plane is beyond our understanding..."
If all else fails, are you guys that right group for him?
You also said that:
He generally seems much kinder to other dms, doesn't have the remarks or throws tantrums to them. (He also is my older brother)
So it could very much be as simple as -Younger brother has to do it my way rawr- or that the tables play-style and campaign just doesn't fit him.
Saying that you play to his level, but then having a level 15 player challenge a lv 20 NPC, says to me that maybe he doesn't want that much of a challenge.
He could be looking for something a bit more cruisy, or maybe he just really really wanted to play pirate captain.
But either way, if you can't find a middle ground with him, maybe have another look to see if he matches your table. It could be as simple as a few small tweaks to the campaign to keep everyone happy.
One of two things is going to have to happen for your group to operate harmoniously. Someone is going to have change how they behave at the table or someone is going to have to leave.
Here are my recommendations for the former. What you've mentioned are actually two distinct (if possibly related) problems. One is that the player is incredibly defensive and the other is that the player is not into PvP in their games.
I'll start with the latter problem because its the easier one to solve. RPGs are a game of cooperative storytelling. That means that everyone has to buy into the premise for it to work. If the game is strongly built around PC conflict, then first you need to make sure she understands the nature of the game. If, after the explanation, she's still not buying in, it may be better to ask her to leave the group. You could still invite her (or everyone) to play a game that is not PvP oriented.
On the other hand, if this is not a PvP oriented game then it may be better to have a conversation with the group about My Guy Syndrome. Explain that there is almost always another choice that makes sense besides attacking the other PCs and unless both players are into PC conflict, it is probably best not to.
If that doesn't resolve the issue on its own, remember that people can't control their feelings (and its generally unhealthy to try) but if they understand their feelings they can better control how they react to them. So my recommendation would be that the next time she starts displaying passive-aggressive behaviors, wait until she is ready to leave and then (depending on circumstances) either walk her outside or ask her to wait behind and talk to her about how she's feeling. Don't judge it; just "You seemed pretty upset during the game. I'd really like to help you enjoy the game. Would you tell me more about how you were feeling and what made you feel that way?" Whatever you do, don't question how she's feeling. Instead try to understand what she's feeling and why. Once you do, you can start asking questions like, "That sounds very frustrating. However, they're not shutting you down because they don't like you. People need to point out when a solution isn't likely to work because if they don't there are usually consequences. But maybe there's another way we can handle that. If you were in their position, how would you handle that problem?"
Best Answer
Talk to your DM about it
From the sounds of things, especially the extra information in the comment to the other answer, you have enjoyed playing with everyone but the extroverted player, and the extroverted player is new to the group. It sounds like the extrovert is not a good fit for your group. From a GM perspective, the best way to handle this is to talk to the player in question to see if they can change their behavior, and to kick them out if that doesn't help.
Making sure everyone has an enjoyable time is one of the duties of the GM of the game. If you have a problem with another player, talk to your GM privately, outside of the game. Tell him that you feel that the new player is not a good fit, and that he is trouncing everyone else. Tell him that you're not enjoying the game as is, and that you feel the new player is the cause of this, and ask him to talk to the new player. Remember that you might not be alone in feeling this way, especially since the other original players are all introverted. Also, show him the next part of this answer.
Advice for your GM
One of your players has a problem with another player. As the GM, it falls on you to resolve conflicts like this. The problem seems to be that you have a group of introverted players, and you introduced an extroverted player who is now taking the focus away from the other players. There is nothing wrong with introducing new players, but it appears that this particular player is not a good fit for the existing group. So, how do you resolve this?
Start off by talking to the new player about this. He is most likely not doing it on purpose, and likely doesn't even notice. Bring it up by saying that "some players" are feeling that he is hogging the spotlight, and ask him to let the other players have spotlight time as well. Make sure that you do not mention who voiced the concern, as this may lead to bad blood between these players. Concerns about other players has to be anonymous. Leave it at that for now, and see how it goes.
I would also make it clear, if you haven't already, that you are willing to listen to feedback. Let them know that they can feel safe voicing concerns to you.
Now, if talking to the new player doesn't help the situation (ask the players who raised concerns!), kick out the new player from the group. The new player is having fun at the expense of your other players, and this is not something you want. If this continues for too long, you risk losing your other players. Losing one player can sometimes lead to a chain reaction, as people realize that they can leave the game at any time.
If this doesn't work, walk away
If your GM doesn't talk to the other player (or remove him), it may be best for you to walk away from this game. This doesn't necessarily mean walking away from the group, though. Tell people that you're not having fun as is, and that you'd prefer not playing in this campaign (if that is actually the case). If you're the only one that feels this way, the problem is solved for the group, and everyone is having fun. If other players feel the same way you do, it is not unlikely that they will walk away from the game as well. If the game collapses, talk to your GM about a new game without the problem player, or even picking up where you left off.
Keep in mind that this last part should only be done if the other methods don't work.