Mostly, it’s because they have to upgrade everything since it’s a new game. But to follow Halo lore, I’ll explain it a little.
In Halo:CE, Halo 2, and Halo 3 you played as John-117, a SPARTAN-II. Without going into any more details, you play as a SPARTAN-III in Halo: Reach.
The IIIs had experimental versions of the MJOLNIR armor and they were able to equip ONI issued power-ups, which explains the jetpacks and such.
The rest of the new Human Weapons are probably prototypes that were lost when Reach fell. The Assault Rifle just has a new look because they could give it one, it’s not really upgraded.
With the covenant weapons, you can just say that they got rid of them, or they just never brought them after the fight on Reach.