Double-jumping in Halo is I think one of the new movement mechanics in H5. Trying to remember from the beta. Same with side-thrust, ground-pound, and the mantle.
I would say Halo's combat sandbox is as rich as you're going to get outside Destiny's sci-fantasy universe of element types and named enemies. But one big thing Halo's sandbox has over Destiny's is the vehicular combat, which has always been a hallmark of Halo's: tightly-controlled, wonderfully-designed vehicles that are fun and easy to use.
Also new in H5 is the ability to command squadmates.
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There is no double jumping
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Woopsie.