1. Foundry - L1, L2, and L3 are all distinct (0, 1, and 2 shooting fireballs), but, other than for the timer (which is important) this plateaus at L3.
2. Barrels - exactly like DK in terms of speed and wild barrel behavior, so you'll need up to L5. The D2K-only triple fireballs start on L2.
3. Refinery - Plateaus at L5, but L1, L2, L3, and L4 are all specific due to fireball speed and the timer.
4. Conveyors - Plateaus at L5.
5. Incinerator - Plateaus at L5, but this board changes the least. The pattern for the "garbage" at the bottom never changes, so as long as you have that down, all you have to deal with are the fireballs. Their speed on L5+ definitely makes a difference though.
6. Springs - You're gonna need a state for every single one of these because the spring movement is level-specific. L14 springs is the D2K kill screen. There's no way to get up the ladder in time.
7. Mixer - L1, L2 and L3 are distinct, but L4+ is the same as L3. In fact, the fireballs even stay at L3 speed.
8. Rivets - Plateaus at L5, as in DK.
Chris pretty much nailed it here but I'll expand on his comments.
Keep in mind that on most screen types which contain fireballs, there will be differences in Levels up to Level 5 due to the fireball speeds.
The "fast" Timer which begins on Level 4 is VERY significant. There are many screen types where you will be in serious danger of running out of time beginning on Level 4.
I'm fairly certain that the "internal difficulty" mechanism which advanced players are familiar with in DK is also in effect in D2K and functions the same way.
Another small detail is that on many screen types there are two different ways that the prizes can be layed out on the screen, and these alternate back and forth each Level. So, in that way, Level 6 could be considered to be slightly different from Level 5, although not necessarily harder.
Foundry -- I'm trying to remember if the fireball which roams the top section ever gets up to Level 5 speeds, or if it has somehow been capped like how it works on the Mixer as Chris mentioned. My feeling is that it DOES, which would make Levels 4 and 5 unique from each other and unique from Level 3 (in addition to the fast Timer beginning at Level 4).
Springs -- Although it's technically correct that they are all unique, I think having a save state for all of them is overkill. I would probably set up a save state for Level 4 and Level 5 Springs since there is a pretty big jump in fireball speeds there which is actually pretty significant to this screen's difficulty. Also, if you care, having these two save states will let you play with both prize positions. The basic approach for ascending Pauline's ladder is the same beginning on Level 4, but your positioning will change based on the difference in the spring's positioning -- again, probably not worth having seperate save states for this. The other thing that eventually gets tough is the jump across the long gap in the top right portion of the screen due to the spring's position creeping further and further to the right -- so, for completeness, you might want to save state one or two high levels like Levels 10 and 11 or something since that jump is very straightforward on Levels 4 and 5.
Well, that's all I can think of to add. It's cool to see some interest in this game.