I figured that posting this in the Strategy Section (apart from the main thread) was a good idea. Not to mention, this is going to get pretty screenshot heavy (I'll post video after I've had a chance to compress and upload to Youtube). I'll be updating bit by bit as I have time to go through and take screenshots from both versions.
All these image were taken before an acceptable method to display full-screen tate on 4:3 tvs was found (set the 360 Display to Anamorphic Widescreen in the main system submenus, then under the Screen option in game, change the Wallpaper to 'Type Clear'). Unfortunately for the PS3 players, this fix only works on the 360 version.
NB: all screenshots were taken as-is from the source video (I did not alter over-scan or perform any resizing). All capturing was done through S-Video at 480i (in tate) on a Hauppauge HD-PVR. The DC is on the left and the 360 version of HD is on the right.
Stage 1:
Stage 2:
Near the beginning of the stage, it is possible to drill the water tower for additional score by targeting one of the trucks parked near the transport ship. In the HD version, the target tracking suffers from an odd glitch where it simply stops targeting altogether (the stream of fire going past the tower doesn't hit anything).

While attacking the pair of docked battleships, the cannon on the bow of the craft remains an active target longer in the HD version than in the NAOMI/DC version. At the point when the screenshot was taken, the cannon has just become inactive in the DC version, but remains active in the HD version for nearly an additional second.

Whilst attacking the first battle ship out at sea, the smaller cannons near the second large cannon can be hit by rocket splash damage from offscreen in the HD version (despite the slightly different times that the shots were taken, this is constant regardless of location). Look at the bonus damage counter.

Same battleship, the bridge becomes an active target sooner in the HD version (even though the hit flash didn't come out in the screen shot, the ricochet flack from the copter's fire can be seen bounding off the tower, demonstrating that it can shot).

During the boss battle, the main bridge remains an active target for the duration of the battle in the HD version. In the NAOMI/DC version, as soon as the first green chopper on the right appears, the tower becomes inactive until the boss timer appears (the explosion in both images is the first chopper on the right being destroyed).

Stage 3:
As the midboss moves onto the screen, each of its smaller swivel cannons begin firing at the player in the NAOMI/DC version. In the HD version, the swivel cannons don't fire until the midboss in completely on screen.

Stage 4:
The formation of 2 heavily armoured tanks, 1 rocket battery and limo that enter from the upper-left near the start of the level begin firing much sooner (in particular, the rocket battery fires well before it's onscreen, making Option Pod deployment difficult). In the 360 shot (which was taken several frames earlier than the DC shot), an enemy rocket can already be seen beneath the remaining lives.

Towards the end of the stage, a formation of 3 small green helicopters descends. In the DC version, they begin firing shortly after entering the active plain. This time has been lengthened in the HD version (in fact, the chopper on the right never gets a chance to fire at me in the HD version as a result).

Just a general example of the difference in bullet pattern density between both versions.

In this final area of 3 cannon turrets (2 small, one large) right before the boss, it's possible to prevent one of the smaller turrets from firing by moving it offscreen in the DC version. In the HD version, the cannon tracks and fires at the player regardless of location.

It's a bit difficult to see in this image, but the reticle lock-on for the smaller cannon tracking during the boss fight is incorrect in the HD version. In the DC version, the player's shots track in a straight line right down to each cannon (as they should). In the HD version, the shots spray wildly, making the cannons more difficult to destroy.

When the boss sweeps its searchlight over the player, the screen goes completely white, save for enemy fire in the DC version. In the HD version, it is still possible to see the player's fire (the thin vertical yellow lines). Also, the main turret's bullets are shaped differently in each version.

There is missing geometry in the HD version. Behind the boss tower are two large defensive walls. The second rear wall is missing in the HD version (there's just a void there which is far more obvious in loop 2 and in New Order Mode in general).

Stage 5:
http://youtu.be/jmoUUJek2ZAThis comparison video demonstrates many of the issues with the HD version that can't be shown in simple screen shots.