I finished it once, couldn't care enough to revisit it after that. I finished it with Flak (the alien guy with the different pets).
I find BL2 to be vastly superior to BL3. I did, however, dig the loot in BL3. Mostly that some loot in later (character) levels and high difficulty settings start to have more interesting stats (finally), but the best part in my book was that weapons stats now can spawn with added points to specific skills. So the potential in build variety is higher in BL3 compared to BL1 or BL2.
But that's it. That's the only positive part of BL3 for me. I didn't like the soundtrack (it's nothing compared to BL2's), didn't like the majority of the characters even including returning BL2 ones. I found Lilith to be very boring in this one (not that she was the best in BL2 but she had more 'punch' and sass to her character there than she does in BL3). Ironically enough, I did like Tannis in BL3 about as much as I did in BL2, very well done character and voiced by the same VA voicing Lilith (Colleen Clinkenbeard ). As a fan of BL2's Tiny Tina I found her to be less memorable in BL3 (I absolutely loved her in her DLC for BL2, amazing work there especially the writing and voice acting). The new young girl (can't even recall her name) that pops up in the story alongside Maya at some point irked me to quite a high level all the way until the end. The "antagonists" twins are a miserable pair that serve more like a cringe comedy duo than 'villains' would. It does "fit" their character to a degree (I.E. they're young, immature, very destabilized kids and act impulsively) but it just wasn't my cup of tea. I mean c'mon, a pair of Sci-Fi streamers are the villains. Like "don't forget to like and subscribe folks!". No. Sorry, but no.
The enemies were boring for the most part, as well. The A.I. is a bit better but ironically it ain't that necessary in this type of game (but it's welcome for some types in particular), even though I could consider it as a minor plus on the side of the bad things in this game. The maps, in particular, or rather their setting and the general environment I thought were pretty boring as well. It's not that they were not well-made, per se. In terms of assets quality, texturing, overall graphics fidelity it's there. It's just that I didn't get 'immersed' much in this game's environments. There were some few exceptions here and there though, some maps were alright; but in general they bored me, especially the first one and the swamp map with the dinosaurs (or whatever those animal names are).
Now, to answer the original question of "does it get better"?
That's obviously very subjective but for me at least the story did have at one particular spot late in the game its sort of semi 'redemption' moment. I do say "semi", because it's very brief but I did like it.
Just in case this may be spoilers territory. So for me that positive part of the story was essentially Typhon DeLeon, the 'First Vault Hunter'. I'm not sure what's the general consensus on him by the community but I for one thought that he was the best new character in the franchise so far. The voice acting is perfect for him, the writing is very good, his actual story (once more is revealed as you learn more of him from his recordings, and himself later on) is actually pretty good. And near the end when I couldn't care less anymore about those two twins and had already turned my brain off for doing whatever it is I was doing in the story just to get done with it this Typhon DeLeon guy gets in the story and turns it around enough for me to wake up again and mentally go "ZZZZzzzZzzz--hu-mmm-oh OH yeah sorry I'm here! I'm here! So, where are we now?". I'm not sure what 'clicked' but he's just intelligently made for the story. The big problems are two fold with him, however: 1) His presence (in person) in the game is WAY too brief for how good he was made, and 2) I could sort of see where he would end in the story but I almost wished there was room for more of his character in a future game or a DLC but... of course... he was just there to help move the story along to the next step, a mere tool.
Now when it comes to pure gameplay alone outside of the story... it depends on a few things. In general, for a BL game it's about two things: Loot, and Playable Characters and what they do and what's their gimmick. For me, even though I do find BL3's loot overall more 'refined' and providing greater build variety potential, the Characters are extraordinarily boring this time around. All of them. The least of which I picked up to finish the game once and that was it. I greatly miss Axton and Gaige from BL2, they were my favorites and basically both were the main reasons why I finished BL2 about 7 times so far; 3 times for each of them, and one time with Zer0. So for BL3 I decided ultimately to stick with Flak and just force myself through the story. Even though I wouldn't go as far as to say that I "enjoyed" Flak, he did have a pet (three choices too) and offered me some respite during fights thanks to the pet actually doing things on his own with its proper A.I. to fight and help me. That's also why I loved Axton in BL2 with the turrets (up to 2 of them with the skills) and Gaige with her awesome robot that moves around and also does things on his own without the need to aim at specific targets or command him to do anything specific if you don't feel like it.
In the end that's the most lacking aspect in gameplay for me in BL3, no passives that help much, no "companion" style tool to help much (outside of Flak's pet) by your side, and coupled with mostly boring maps, cringe antagonists and a story that I couldn't care about and was terribly paced on top of it... yeah, but... no I suppose that the final answer should be that the game "didn't get better" overall, save for some very specific and isolated 'better parts' near the end, it wasn't enough to exclaim at any point something like "Finally! I'm actually starting to really like this game now!". Nope. I didn't want BL3 to end up like this for me but it did. Then again, no biggie, I still have BL2 which is a masterpiece in comparison to BL3 in my book.
Now I don't "hate" this game. On its own, I guess, it's 'alright'. I'd still give it something like a 6.5/10, it's passable for one playthrough; then move along and forget. It's what it did for me anyway.