Yes, I very much agree with Walter. It's even said early in the (anime) series:
"To protect his little bruised heart,
One grabs a sword.
To forge his own destiny,
One grabs a sword."
-Clearly these descibe the paths of Gatsu and Griffith. Gatsu grabs the "sword" because he has to. Griffith does because he wants to rule the world. Though we never really gain so much insight as to why Griffith has that particular dream. I may have just missed it, but isn't that right? I mean we don't know any backstory on Griffith, other than from Caska's flashback.
There are many unanswered questions with Griffith. But I belieive that they will be answered, eventually. Griffith is a character shrouded in msytery. Even the members of that were closest to him didn't know him that well. One thing that was apparent, though, is that Gatsu was a big part of Griffith's dream. Griffith needed Gatsu by his side. Why? Because Griffith had feelings for Gatsu. Feelings that even he would not acknowledge. Gatsu was Griffith's prize, and as much as Griffith tried to deny his feelings he could not stop them from coming after Gatsu left. Gatsu ultimately made Griffith betray his dream. Gatsu made Griffith feel helpless. Now I think Griffith's dream is not only to try a rule the world, but to incorporate Gatsu into his life again. To "win" Gatsu back by making him suffer. And in many ways, I think that dream supercedes becoming King.
As for Gatsu, the only dream I think he has right now is for things to be more like they used to be. No monsters, no insane Caska, just a normal life. I think that's all Gatsu has ever wanted. He has always struggled to be accepted. When he finally realized that the Hawks accepted him, even admired him, that was his dream: to stay with the Hawks, and Caska. But Griffith crushed that dream. One thing I would be interested to find out, if Miura ever chooses to reveal it, is Gatsu's father and mother. Why was she hanged? Who was his father? I think Gatsu unknowingly seeks such answers, and his dreams would radically change if he ever found out.
It's quite ironic that Gatsu and Griffith destroy each other's dream...but then again one does not pick up a sword in the name of his dream unless he is prepared to strike another down.