I should not have gotten myself into a position where I can write this. I should have just gotten a Professor Layton cartridge from my sister; I know she has one. But I made my choices, and now I'm going to document the results in agonizing detail.
Plot: if you liked the plot of Puzzle Agent, the beginning and end of PA2 are okay, but in the middle, the puzzles are filler between the incoherent plot, and the plot is filler between the incoherent puzzles. It is like an Ouroboros of suck, except I might look for porn of that, so it is therefore worse than an Ouroboros of suck.
Anyway, that's the plot, the style is the same, so... puzzles. I'm going to try to break this down by category, and judge each puzzle by its merits according to the conventions of its category.
Puzzles that I would argue are not actually puzzles:
Moon Focus: On the one hand, it did not actively offend my puzzle-related sensibilities, on the other, it's not terribly obvious why a telescope would do that. Let's move on.
Cross the Streams: I am only dignifying this puzzle with a sentence to communicate my desire not to do so.
'Round The Cosmos: It looks nice, but, well... click click click, click click click, click click click, and you're done.
Rotating tile puzzles: these are more a subset of the previous category, but I like them because they offer an outlet for my obsessive tendencies. Still, I'd recommend an actual cardboard puzzle. More expressive power, better bang for your buck
Retrace the path: the most sophisticated puzzle of its type in this game, which says some pretty terrible things.
Starry Field: Glorious, glorious obsessiveness, and the art's good, too. Not much to say.
Fill in the sequence puzzles:
Monosaki Puzzle #320: This is not a puzzle. This is the result of someone wanting an easy puzzle and staring at a clock for too long. I actually blew a hint on this because there is nothing puzzle-like about the obvious (and, insultingly, correct) answer, and the hint it gave me was itself, preposterous.
Coins in a Box: If you're not a nerd with a knowledge of US currency, this will be a pain. Fortunately, any non-nerds will have decided that this game is a waste of their time long before they reach this puzzle, so effectively the only requirement is knowledge of US currency.
Digits In Space: Telltale. Do you see what you have done here? Do you? Never do it again.
Monosaki Puzzle #512: This is the best sequence puzzle. It should have been the bare minimum of quality.
Puzzles wherein you are required to arrange things according to some arbitrary conditions: (this probably has a shorter name)
Finding Vacancy: Adequate. Somewhat on the low end, but it is early in the game
Missing Persons: The fact that the letters marked form something that looks not terribly like a word threw me off. Also, Nelson Tethers is fucking psychic to figure out what the puzzle was supposed to be. On the other hand, he couldn't intuit the answer, which would have bypassed a fair number of the other puzzles. C'est la vie.
Find Edvard!: There's probably a better category for this, but I'll leave it here for now. This was all right, even if it did drag on for... quite a while.
Lunacy Algorithm: Like Find Edvard!, in terms of the overall experience, except over quicker.
Arrange pictures in chronological order:
Hidden Sightings: Very blah. It is a chronological order puzzle. The game has both better and worse.
Clearing The Ice: This is the better. It's got a twist on the formula that made it require actual thought.
Gnome Appearance: Remember Thief of Time? That was a good book. I really appreciate Pratchett's gift for sounding plausible when he explains the problem. Writing a world where stuff is based on plausibility was a good choice for him, obviously. As I recall, the clock was harmful because it tried to divide up time too finely. The characters likened it to trying to open a box using the crowbar inside the box. ... What was I saying? Oh, yeah, solving this puzzle without hints requires a plot point that is revealed by solving the puzzle. Weird, huh?
Raccoon Spa: This is an optional puzzle you can get by clicking on the fourth Hidden Person after you summon them to Olav's cabin. Don't click on the fourth Hidden Person. Just... don't. This puzzle had potential, but it just doesn't present itself right. YMMV, but since it's an optional puzzle, I'd say leave it to the end, anyway.
Puzzles involving dividing up multiple sources between multiple sinks:
Stacks of flyers: NO. NO NO NO NO NO NO NO NON ON ONO NONON ON ON ONO NO NO N ON ON NO. My anger at this is inspired by a misreading of the rules, but it's a sensible misreading that a sane person would make. "We need to distribute 50 flyers." "Is a 1/3 chance of 51 okay?" "No, and you are a terrible person for thinking so."
I said I misread the rules. I misread them because they don't match the dialogue, and I went with the dialogue's presentation of this puzzle, which the game does not actually accept. Ugh.
Power Grid: Meh, whatever. I will note that the flexibility the solver checker allows means that my solution does not actually match up with their explanation in any meaningful fashion, but at least it took it.
My Four Moons: This makes no physical sense on a surprising number of levels, and it's not really great as a puzzle, either.
My Three Suns: Everything I said back there applies here, except moreso.
Cut the obstructions with one cut for some reason: I don't get it. Does Nelson Tethers lug around the Masamune or something, but he's only worthy to use it once at any given time?
Sheriff's Door: I got thrown off by this puzzle's 'twist', but I think it was too little, too late. YMMV, pretty clearly in this case.
Olav's Cabin: Can I get a meh?
Sliding block puzzles: I don't know if this was just me, but it seemed like these all had kind of a terrible interface
Sheriff's Office: Pretty okay. I think this might have been the best of the bunch.
Earth, Sun, & Moon: You know what sliding block puzzles need more of? Visual noise focusing on the shapes and boundaries. Wait. NO THEY FUCKING DON'T.
Finding Alfred: Meh.
The Escape!: This puzzle was complicated by my inability to realize that Nelson's head takes up no physical space. They should have said that in the puzzle description. It would have been right in line with their standards for puzzle-writing.
Reflect light from a source to a destination: the mechanics are improved from PA1. Besides that, I don't think there's much to say about these puzzles. Besides the fact that Nelson Tethers can apparently only rotate things in puzzles focused on rotating things.
Bjorn's Sanity: Yep, this was definitely a reflecting light puzzle. That it was.
Solar Rays: So was this. Yep.
Puzzles based on predefined grid-based movement (mostly programming puzzles):
Mission: Rock: The boundary of this puzzle is not trees. It is water. That should be all you need to know to solve it the first time, and the fact that I felt the need to point this out should be all you need to know about my opinion of this puzzle.
Lander Circuitry: A G-Man just pulled a gun on me! He's taking me away from the puzzle! Hooray! *later* Wait, I have to do this anyway? Rats.
Robot Hero: I cannot muster the will to think or talk about this puzzle. It is optional, and appears to have been thrown in for the heck of it.
Super Robot Hero: Robot Hero, but, um, moreso.
Anyway, now that I've laid out my opinion on all of that... Shame on me. I got fooled twice.