*whistles* Just read through this entire topic, and there are some very good points and theories...
I agree mostly with Meggiemacat, iLLViLLAiN, and Saphira. I'll elaborate on my own opinions and hopefully you'll see why. ^^
I believe that NiGHTS and Reala can indeed be considered "brothers", in the sense that they were both created at the same exclusive level and assumedly around the same time, causing a closeness by default. Heck, they even refer to each other as "siblings" in the game.
I also believe that, in the beginning, they were both completely loyal to their father and really lived as true Nightmaren-- ferocious, frightening, and all that.
However, both NiGHTS and Reala are "equal in ability" (although NiGHTS is later stated as being somewhat superior to Reala in this respect) but of "opposing character", as it says right in the JP manual. Reala is naturally cruel and brutal, and remains absolutely loyal to Wizeman even today.
NiGHTS has a bit of a personality problem, though. Once again, the Japanese manual states clearly that NiGHTS has a "free, wild character" along with the "craziness of the original Nightmaren." He doesn't like restriction, doesn't enjoy being told what to do and who to follow, and originally seemed to want nothing more than freedom from his father's power by any means.
Cue the manual again, it mentions that "the free, wild character of this individual suffers much misfortune." NiGHTS is also not only the most gifted but the most powerful Nightmaren that Wizeman created. I assume that former statement is hinting that NiGHTS literally felt imprisoned by this role as 'daddy's favorite', despite the prestige and power, as he constantly had to live up to expectations and guidelines. Undoubtedly that was the catalyst for his imminent mutiny, and a huge factor in why he still won't go back today.
Note this, though, and let's get back on topic-- it is stated as canon information that Reala was responsible for NiGHTS' imprisonment, although the actual confinement was Wizeman's doing. Reala undoubtedly envies NiGHTS bitterly simply for his superiority and rank, and is both understandably hurt and furious because he so carelessly tossed it all aside for personal interests. Reala would kill for NiGHTS' position, and this absolute refusal of it by the purple 'Maren-- along with his tendency to openly and frequently oppose his own father-- enraged Reala enough for him to literally disown his brother ("There is no NiGHTS!") and join Wizeman in throwing him out of Nightmare.
However, I feel I must add a little more on NiGHTS' and Reala's current situations with their recent JoD actions. It has been mentioned several times in this topic that NiGHTS apparently displays concern for his brother, and vice versa. Well, geez, it's been over ten years since his initial rebellion, maybe they've all mellowed out somewhat? (Except for Wizeman, of course-- he's become much worse and much angrier...)
Also, even though NiGHTS may have originally aided all 4 dreamers with a large amount of selfish interest, he truly seems to develop some real concern for their own well-being over time. He may always be a nasty Level 1 at heart, but NiGHTS has definitely learned how to be kind and caring, has learned how to think of others rather than himself. Reala hasn't learned this yet, simply because he hasn't had much chance and because he's naturally a nasty guy. There is some evidence of more positive qualities in him now, though... you all mentioned his nonviolent dialogues with his brother, and the fact that he gives him many chances to change his mind and return to Wizeman instead of just dragging him back there by force, although he has indeed threatened to do so.
That previous paragraph, though, just adds more fuel to the fire on the vast differences between NiGHTS and Reala's personalities. NiGHTS is playful, carefree, and hates any sort of restriction. Reala is all business, takes his work very seriously, and lives by the rules his father has set. The two just don't work well together, at all.
Personally, I don't think NiGHTS and Reala can ever truly hate each other, although they sure as heck can seem like it from time to time!
One thing is true, though, and that is that they will always be bitter rivals. That's a given.
So there you go. Opinion, explanation, and backing proof. Sibling rivalry to the extreme.
Sorry about the ludicrous amount of words there! I tend to elaborate a little much...