Personally I feel like part II is a much better game.
Gameplay:
Both games have the same general structure as a third person shooter with stealth mechanics. The first game especially is pretty basic, and it's a product of last gen where TPS were very popular and all over the place.
Both games are pretty much completely linear as you go from point A to point B and but in part II, although it's very much linear, you sometimes get this feeling of faux freedom in the way you're going so it can keep things a bit more fresh and the game overall feels less claustrophobic and handholding.
The game feels much better to play in part II though and I doubt anyone here will think otherwise. Ellie & Abby are both much more agile than Joel ever was, and the dodging, jumping, prone, and other additions to the gameplay overall make you more engaged in the combat, therefore making it more intense as well.
The shooting feels much better, to no surprise. That and also the whole gore & blood system ND made in part II is absolutely nuts and the violence feels absolutely ruthless, unforgiving and cruel. Enemies beg for their lives, shout their friends names when they die, become angry/sad, etc. It makes everything feel more alive, and like you're actually fighting people in the LoU universe as opposed to just your regular ol' NPC enemies. The shotgun in part II is probably one of the most satisfying weapons I've played with, that thing just tears people and infected apart.
Another huge plus to part II is the removal of those annoying dumpster/ladder "puzzles" which were all over the place in the first game. After a handful of those I literally groaned every time I was forced to do another one. You have them in part II like.. twice but I think they're more of a nod to the first game.
Story:
The first game's plot to me is a standard apocalypse story with an emotional detached tough character and a young cool one that the tough one learns to love and heals him. It's not original or new by any means but it's done well especially because the characters themselves are nicely written and acted. The story didn't really click with me until after Joel gets hurt and Ellie takes care of him. From that point I was more invested in the story but overall I don't think it reaches any of part II highs by a longshot.
This is the 'controversial' part of part II. I feel like the vast majority of players actually liked and respected the story but the whole leaks fiasco and the vocal majority lead people to think otherwise. Sure, the game goes from 0 to 100 straight away and doesn't shy away from making hard decisions.
The thing many people were angry about happens pretty quickly, in order to drive the plot forward. I think ND should be applauded for doing things in an unorthodox & uncompromising manner. They had their vision of how the game should and needs to be and they did it, because although the players are attached to Joel - in the universe of the game he's a mass murderer who may have sacirificed humanity's hope for Ellie, and his own sake. He is no saint and his death was coming sooner or later.
The writing, characters, voice acting, cut-scenes - everything is leagues beyond the first games and sometimes I felt like I'm watching an HBO show and I was a lot more invested in this game. I think the first game's plot is a bit overrated and it didn't really kick of for me until more than halfway through the game.
Here? Right from the beginning I was invested and besides having fun playing the game, I honestly wanted to know how things progress and what comes next.
The story can be a bit confusing at first with its non-linear way of story telling like how it incorporates flashbacks and jumps off between Ellie's & Abby's days in Seattle but I liked it and felt like it was building towards something great which eventually was proven to be the case. The ending of the game is emotionally difficult (at least for me) and also very powerful.
The game managed to introduce a new character that kills of our beloved Joel straight away, and we as the players are going to hate Abby right away and wanting to take our revenge on her, reflecting Ellie's feelings. Fast forward a dozen hours or so - we take control of Abby after a very intense cliffhanger which I bet made many people upset but it's another decision by ND that I respect. By the time Abby's playthrough came to an end I liked her as much as I liked Ellie and I was sympathizing with her and her actions. The last part of the game forces you, as Ellie, to fight Abby in order to take your long sought revenge and I honestly did not want to go through with that fight and I was disgusted by Ellie's actions. Thinking that just ~20 hours before that I was introduced to the character and hated her guts is pretty amazing and quite an achievement by ND. Nothing in part I made me feel nearly as much as part II and its incredibly powerful & emotional ending and message.
I haven't got much to say about the sound & graphics, not because there's nothing to say but because they are both incredible and everyone will agree on that. Pretty much the most impressive PS4 game to date.