1) The present continuous is often used for arrangements, and when we have prepared for the event somehow. So here it emphasizes that my brother and I are both prepared to meet, we have already talked about it and decided, maybe made particular plans for that day.
In another example, if you say "I'm flying to Paris tomorrow.", it means you have bought a ticket, taken time off work, etc.
2) Future with 'be going to' is usually used for plans and intentions. So if you say that you are going to meet your brother, you intend to do it, but you might not have taken the necessary steps for this to happen.
You can say something like "I am going to retire at the age of 60, buy a house and plant a cherry tree in my backyard." You might be 30 when you say that, it shows that you have thought about this and think of doing this, not that you have already bought a house. But if you say "I am retiring next year", that is pretty fixed.
The two are really close and often interchangeable. In the examples you are giving, I do not see a reason to prefer one over the other.