Might do a run on this game, but id like to know the difference between all of the missions category’s in the levels leaderboard

Mission 1 is the regular mission you'd do when playing through the story.

Mission 2 is the "Collect 100 Rings" mission.

Mission 3 (except for Route 101/280) is the "Find the Lost Chao" mission.

Mission 4 is the regular mission, but with a time limit and without checkpoints.

Mission 5 is the "Hard Mode" mission, which just has different object and enemy placements from Mission 1/4.

if you mean mission 1 through 5, that's just the normal missions 1 through 5 each stage has in the game

if you're asking about the catex

Story Style is doing an M1 run with the upgrades you would have access to at that time. Like Prison Lane without the hover.

Restricted are M1 runs that follow the Restricted ruleset. (https://pastebin.com/6asrb2bm)

M1/M4/M5 A-Ranks are ils for those respective missions that route in points to get an A rank.