Just got Trine 2 in the recent sale, and it's absolutely gorgeous.

What would I be missing if I didn't pick up Trine 1?

Differences between Trine 1 and Trine 2:

Visual Difference:
Trine 1: sharp clean graphics, subtle fantasy lighting.
Trine 2: more saturated color, glowing effects, brighter color, edges are softer

Hud:
Trine 1: No option to disable the hud (you can disable it using a convoluted method I've posted here:
http://hudfreegaming.wikia.com/wiki/Trine
Trine 2: You can turn the hud off in the options

Character changes:
Trine 1: Knight can pick up and throw objects
Trine 2: Knight can no longer pick up and throw objects

Trine 1: Wizard has to pick up mana to perform magic.
Trine 2: Wizard does not need mana.

Trine 2: Some new character abilities were added (example: the knight can use his shield like a hang glider)

Controls:
Both games play great with an Xbox controller (I recommend using a controller).
Trine 1: setting up controls for the keyboard and mouse can sometimes be buggy.
Trine 2: I recall controls being streamlined and easier to set up.

Gameplay:
Trine 1: Most puzzles can be solved using any of the characters. It's even possible to beat a whole level with only one character (great challenge).
Trine 2: You must switch characters to progress. So, puzzles feel more rigid in how they can be solved.

Combat:
Trine 1: Combat is slower and each of the characters can fight (although the wizard can only draw planks and boxes to drop on enemies).
Trine 2: Combat is much faster and the knight is most useful here. The thief is ok at range. The wizard can use magic to trap and pick up enemies now. However, he often doesn't have time before he's killed.

I wish the wizard could shoot fireballs, but he can't in either game. Oh well.

Extras:
Trine 1: Finding hard to reach secrets rewards you with items that effect gameplay.
Trine 2: Finding hard to reach secrets just unlocks concept art and poems.

Levels:
Trine obviously contains a lot of levels that aren't in Trine 2. So, if you don't buy it you'll miss out on all of those levels.

Music:
Trine 1: has awesome music and you get the soundtrack when you buy it on GOG (that's worth the price alone in my opinion)
Trine 2: Good music, but not as good as the music from Trine. Soundtrack doesn't come with GOG version.

Personally, I like the levels, art, gameplay, combat, characters, voice over work, and music in Trine more than Trine 2. I like the hud options in Trine 2 more and it's a little easier to pick up and play. Bother are still great though.

I was not expecting such a thorough response! Thanks so much! I'll definitely be picking up Trine at some point!

I'm in the same boat, although I haven't fired it up yet. I'm looking forward to it though. If I like it I'll probably pick up the first one at some point (in part based on this thread).

UPDATE:
I got the change to run through the first few minutes of the game, and I think it's amazing. If it lives up to the potential I see here, I'll definitely get the first one too, and I'll probably recommend these games to my friends who have kids.

With the recent announcement of Trine Enchanted Edition I was wondering if there would be any benefit to the engine switch and decided to look for opinions on the differences. Your post covered everything in one fell swoop, thank you so much for taking the time to compile the list. *thumbs up*

Can't really see why they decided to make that new edition, Trine is almost perfect as it is. No harm done, of course. It seems like wasted effort though.

The main differences between Trine 1 and Trine Enchanted Edition seem to be:

1. Trine Enchanted Edition will use the rendering engine and lighting of Trine 2. I think that's sad. I prefer the less saturated more realistic lighting and color tones of the original Trine.
2. Multiplayer support.
3. The ability to turn the hud on / off like you can in Trine 2. That option was not in the original version of Trine.

So much this. I think Trine has the perfect balance between fairy tale and realistic aesthetics that's very rare to come across. They went a bit overboard in T2.

Anyway, it's a good thing (if I understood correctly) that one can choose between original and Enchanted Edition without much hassle, so nothing is forced. :)

Kudos to Frozenbyte of course for still supporting a 5 year old game.

The multiplayer support on the GOG version doesn't exist. In other words, local co-op is the only option, like it was before the Enchanted Edition upgrade. This is in contrast to the Steam version, which has the advertised online co-op.

This is nothing more than a blatant lie. Have no idea where you've heard that.
These images come from the Trine Enchanted Edition downloaded from GOG just now.

If that's true, the store page for Trine Enchanted Edition lies to consumers then. The store page says the following:
Multiplayer notice: Trine supports mutliplayer via hotseat. Online or LAN play are not available.

Oh, yeah. That's probably the old info which was not removed yet. I'll try to tell the support about that.

Nice :)