I love both games a great deal, but what makes me love 1 more than 2 is a few things, actually, most of which are related to the gameplay:
In Nights of Azure 1, the world is split into two parts: the city area and the dungeons. And the city area connects to all of the dungeons and back to the hub (of course, once you are in a dungeon, you cannot leave until you beat the boss or find a warp gate to take you back to the hub, because they are one-way only). Nights of Azure 2 doesn't have a connecting area like that, and while the dungeons are a bit longer and not just one-way, I prefer the way Nights of Azure does it, because it makes Ruswal Island (where NoA1 takes place) feel like an actual world, and you have a sense of where everything is, which is kind of missing in 2.
Nights of Azure 1 does not have a time limit mechanic that makes it impossible to get everything on a first playthrough (which is sadly the case in 2).
Nights of Azure 1 has the more memorable boss fights (with the notable exception of the final boss which I vastly prefer in 2, though 1's is also excellent).
Arnice feels better to control in combat. Her movement is more fluid. And I prefer Nights of Azure 1's weapon switch system to Nights of Azure 2's, where different weapons are tied to different Servans that can transform into them.
Speaking of the Servans, Nights of Azure 1 has the better Servan system. Not just by virtue of having more of them, they also feel very much different from each other in gameplay; a huge part of NoA1 is finding a good Servan party to aid you in fights, and while the game is perfectly beatable with just the ones the game gives you in the beginning, some of the bosses and arena challenges are much harder than they would otherwise be. In addition, you can have up to four different parties of Servans and you can freely switch between them as the situation demands it.
Not to mention, while it is neat that NoA2 gave some Servans abilities that could open up optional paths to treasure and more Servans, this means you won't really be using any of the Servans without these abilities if you want all the optional stuff. You can also only ever bring 2 Servans and one human partner (the human partner system isn't in NoA1, to be fair, and I liked the addition of it), making it feel a lot more restrictive.
Now, some non-gameplay stuff:
Nights of Azure 2 has Lucia, my all time favorite theme in video games (heck, favorite song in general), but overall, I find Nights of Azure 1's soundtrack to be better. Not that NoA2's soundtrack is bad, far from it, I just prefer the first game's.
Nights of Azure 2 tried to add some variety by introducing playable side characters (which constitute the human partners). However, after their introduction, they sadly don't matter much in the grand theme of things outside of their own character side quests.
Nights of Azure 1 is very much the story of Arnice and Lilysse, and it makes that clear from the beginning, but it does a better job at keeping its side characters relevant even without them being playable.
While I really like Aluche, I find Arnice to be the better protagonist overall (well, she IS my favorite protagonist in gaming, so...).
On PS4 at least, Nights of Azure 1 runs a lot better. 2 has some framerate issues, and in general, it looks like it had a smaller budget behind it than the original game.
This is not exactly a dealbreaker, but I found it worth mentioning nonetheless.
Now, again, I love both games to death, Nights of Azure 2 is still one of my top ten favorite games ever, but I think this about covers it.
Now some things I find NoA2 does better, so that it doesn't look like too bad a game:
The antagonist and final boss are much improved from the first game
Nights of Azure 2 has selectable difficulties
The Servans are predetermined and all you need to do to get them is to find them in the world;
getting the items to summon Servans in Nights of Azure 1 can be quite a hassle, although once you have the item, you WILL get the Servan associated with it and you can have multiple of the same Servan. I still prefer 2's method in this regard, however. Servans in Nights of Azure 2 have more fleshed out personalities because of this (like the boomer dinosaur, for example). In Nights of Azure 1, each "type" of Servan has the same personality and voice (Fay-type Servans are all pranksters, for example).
While I prefer the way Nights of Azure 1 presents the world to you (as seen above), I do like the dungeons in 2 more. They are longer and have more optional stuff to find and discover, whereas Nights of Azure 1's dungeons are for the most part admittedly linear. Not to mention, you can actually go back and forth in areas and even unlock warp gates to use as shortcuts (as their function isn't just to take you back home anymore).
In Nights of Azure 2, the enemies that appear in the different areas will change depending on how far you are in the story. In Nights of Azure 1, they will always stay the same for the most part, with some exceptions being areas you have to revisit in the postgame.
Phew. Sorry for the wall of text. XP
