TLDR: I loved Borderlands two and I'm going to start a replay with a different character, which i almost never do.
I had fun with Borderlands 1, but was basically pushing through by the end trying not to fall asleep.
B2, on the other hand, I was very engaged in the entire ride through, although it seemed well balanced and the game ended when it should, even including the side quests.
Improvements -
The writing! I think I heard a reference to Anthony Burch writing in Borderlands 2 from hey ass. Whatcha playing episode, and he got some award?
Experience tiers, which I didn't initially like because it wasn't explained, but basically the quests are more important than farming enemies for experience, so you get much more engaged with the stories because you're following quests instead of trying to kill enough boring enemies in the same way to get high enough of a level to destroy future enemies.
In any case, the writing that I did like in Borderlands 1, was perfect in 2. Not too long, always funny, always engaging, every character very well defined, which brings me to my next point:
The voice acting was hilarious and perfect, again, very well defined and idiosyncratic for every character, just so much fun to listen to. Every time handsome Jack pops up. It's fun to listen to him be an a******.
The quests were so much more satisfying. At the end of the first game, I was basically just following marker to marker without caring what anybody said or what was written down just tapping through to the next wavepoint until the quest was finished.
In Borderlands 2, Even if I accidentally clicked through the introduction to the quest, I would go back and make sure to read because I know that the paragraph introducing and explaining the quest is funny and that the quest is going to be rescuing lab experiments and I have to find a particular valve or putting together a treasure map with a weird lure, rather than just find the bigger bad guy punch him to death.
Driving was huge - I was not into driving in Borderlands 1 and got really bored and irritated every time I had to drive. I felt like the aiming system was complete dog s***, and it was just not very fun to drive around in general, like the handling was terrible.
I loved driving around the Borderlands too and was actively bummed out whenever. I didn't get a car, but it made perfect sense and they used the car just enough so that the game wasn't too easy. But you could still boost and race around however you wanted, or chase down a beer van. So much fun driving, such a huge improvement from the first game.
Larger levels with more interesting landscapes in them, each level felt much more unique than the entire The first game to me, like each area had its own style to a degree I hadn't seen before.
I can't remember a single place from the first game, but I'm going to remember the different style of the underground bug. Bunker and the dust and all these other places that had an impact on the personality of the game.
Lastly, art is more efficient, not as many bold lines emphasizing the comic book quality, which is carried through more by the personality of the game instead of by the specific art.