The trailer of Halo 4 shows the Chief waking up in a stasis pod and he has upgrades to his armour (a slightly different look and a jet pack) and yet he wakes up in the same ship that was 

Also, I'm not trying to troll or flame, so keep the hate mail to a minimum.

It's a sequel. Shit needs to be more awesome. How does that make sense in the continuity?

Sequel.

I would think that Chief having the new armor in the reveal trailer was done simply to show off that he would be having some upgrades to his look in the game; He'll probably start the game off in his Halo 3 getup and eventually get the upgrades seen in the trailer soon after he crashes/lands/is captured. 

Yeah, I don't really care if his armor is slightly different- they never explained his armor's style shift from 2 to 3, either, so it doesn't really bother me

Also with the ship being all blown up there was fire in space. So there's that.

It's only armor man. People analyse things way too much these days.

Halo trailers have always been sort of bizarre and anachronistic within their own fiction.  They also very rarely portray exactly how things happen in the game.

Chief plugged Cortana into that stand/the ship's systems at the end of 3, telling her 'wake me when you need me'. So that part is at least fine for continuity's sake. 
 
The new armour? Eh. Sequel. 343 wanted to put their own little spin on the chief's armour design, it's not really something to get caught on. Like others of mentioned, maybe it's an upgraded spartan armour he gets/finds over the course of the game? I wouldn't be bothered if they don't explain the chief's new suit.

There was slight changes in the armour from Halo 2 to 3 too.

@TEHMAXXORZ said:

Also, I'm not trying to troll or flame, so keep the hate mail to a minimum.

WHAT THE FUCK ARE YOU DOING TO THE BEST GAME EVER I'M GOING TO ASSSAUUULLLTTTTTT YOOUUUUUUUUUU NOOOOOOOOOOOOOOOOOOO!

This is actually really interesting to point out. I didn't notice that Master Chief's armor was any different, but I did think about how he would have added the jet pack before waking up from the pod. I wonder if the scene from the teaser trailer is actually going to factor into the final game at all.

I don't care if they change the armour or even the look of the game. It's a new trilogy and they should change as much as they like while keeping the game feeling like Halo. Now that isn't to say I think it makes any sense or what they are doing is even a good idea. It's going to be a bit of stretch to accept a new enemy that no one has ever mentioned and even worse that they are going back and inserting references to them into Halo 1.

I don't think we know that that was in fact the same ship. I could be wrong, but I don't think it was explicit. Yes, it probably was, and maybe it was confirmed, but it's always possible.

That said, there could easily be explanations. First of all, maybe he's codpeice got blasted off? Second of all, maybe the jet pack is something in the game that he finds early on. Thir of all, it's a teaser trailer for a game that's probably VERY early on in development, and Bungie was always putting stuff in their trailers that had nothing to do with how Halo would actually go. So it's possible that a bunch of stuff leads up to that stuff, or it's possible that he hasn't got a clue why his armor is different either, and there's some mysterious presence on the ship or something. Who knows.

A space-wizard granted him the power of flight and then the alienz attacked so he shoots them good.

That bothered me as well, but I just figured "it's a sequel". They feel the need to throw new stuff on him, and it's pretty minor as far as continuity goes, so I'm willing to look past it so long as they do cool with it.

There are little thrusters in the newer versions of the Mjölnir armour, they are only useful in zero g though.
Cortana is fine, inserted into one of the consoles on the Forward Unto Dawn, which whilst fucked up, was not destroyed, only cut in half. There is plenty of backup power in a damaged frigate of that type to sustain a single AI and a single cryo tube.
343 have mucked around with the look of the Chief's armour, yes, but it's really the same as it was when he went into the stasis pod, he's not been drifting around at all, he's quite secure in there and in fact can not remove himself from stasis without outside intervention. They probably just needed a cool way to get him out of the ship.

One thing I noticed, which no one else seems to have done, is the pistol type weapon that he fires. It's a very strange indeed, like it's a break action weapon, firing some weird grenade-like projectile. Upon further inspection it seems like it's a single use grenade launched from on top of a new/weird pistol.