Doing It for the Art: The reason the game took the controversial route of being so different from Trigger is that the developers felt there was no point in making the same game twice.
Also, the decision to hire Yasunori Mitsuda (who had just left Square) to compose the game and relatively-unknown singer Noriko Mitose (as opposed to a more popular singer) to write and sing the ending song were made against the wishes of Squaresoft PR, simply because their sound was deemed "right" for the game.
I Knew It: This game was created because of one persistent nerd rumor: Schala Lives! ...as three different people at the same time. There's the "original" Schala stuck in the Time Devourer; Kid, Schala's "clone-daughter"; and Harle, who is an echo of Kid. Oh, and at the end, you get to free the original Schala, assuming you paid attention during the last dungeon.
Jossed: The new ending and storyline in the DS version of Trigger jossed some fan theories very, very hard.
No Export for You: Chrono Cross was never released outside of Japan and North America, which makes the continuity nods in the DS remake of Chrono Trigger just plain annoying for everyone living outside those two particular regions. The more general references to Radical Dreamers are also beguiling for everyone outside of Japan, and probably even many people in it.
According to the developer's room, Zoah's helmet hides his secret identity as a prince of a certain country to the north who's watching over the party. Unfortunately, the scene that's supposed to hint at his true origins was never implemented.
Like many PS1 RPGs, there is a huge Info Dump near the end of the game as the writers tried to get their plot points across before the Cosmic Deadline hit.