The Raid is where mechanics change up heavily, including "Destroy the CPU's Cores!" Sections, "Guard the Flag" Sections, "Metal Gear Solid" Sections (complete with Crouch walking/not jumping being the smart thing), Megaman Block Puzzles, and a Shield that makes for great lemonade.
Otherwise missions remain fairly standard. Bungie has heard complaints and aims to fix those, but admittably the stories all tend to be varied and repeating the same strategy every time won't always work.
As far as story: If you actually CARE about story you got to remember this game's only got chapter 1 out. It's meant to be a story put over 10 years of development. This means they couldn't reveal much, the fact you kill a
says a lot about how far in the story we do get, even if it's little in the bigger picture.
The game's currently focused on gathering info on the enemies and stopping a few low scale but major threats they are setting against humanity. Examples include the Chamber of Night ritual on a certain object, the Nexus Strike where you are halting the Vexification of a planet, and the Black Garden's agenda.
As far as gathering info, you got the "Last Array" where you learn of remaining colonies (it seems) of humanity in space, "The World's Grave" where you steal Hive knowledge, "The Archives" where you leech a lot of lost knowledge from the Golden Age, and basically all the Mars missions that are non-vex related as we never REALLY set against the Cabal excluding the strikes.
Deeper story (basically the 'common knowledge' of the universe, with a number of deeper tidbits) are restricted to the grimoire/codex on the site. Mostly because honestly, you got two options: A story with hours of cutscenes to exposition everything, or a story with little in it so people can enjoy their action. They found the mid point.