How do Chrono Trigger’s SNES and DS versions differ

In Wait mode if you're in a command menu (i.e. choosing an item or tech to use) time effectively stops. In active mode, Active Time Battle bar is always going; that's the one that fills up and your character can act when it's full. This includes yours and the enemy's which you can't see.

When you're just in the main combat screen and not in a particular menu there is no difference.

Once you learn dual-techs you might actually prefer Active mode, because in Wait mode you would have to sit in the main screen waiting for both characters to be ready to act, whereas in Active mode you can just select the tech while waiting and then just choose it.

That's actually one of the reasons that Wait mode doesn't make it completely "turn-based" the way you expect it to. You sometimes need to wait till 2 characters are ready to act simultaneously and if time stopped when it was 1 character's turn, you wouldn't be able to do so.

Technically speaking, that'd be the Dreamseeker. Obtained after defeating the Dream Devourer, it boasts an attack power of 240 (to Rainbows 220), and crit rate of 90% (Rainbow's crit chance is 70%). In order to obtain it, you'll need to defeat the Dream Devourer, let the credits roll, and create a new save file. When you start your next New Game+, Chrono will have the weapon.

Practically however, since Attack Power caps at 255, you may find other weapons with different effects to be more desirable once you're at a level of power at which you can acquire the Dreamseeker in the first place.