Let me preface this by saying that I absolutely adore Borderlands, it's one of my favorite series of all time. I don't want anyone to think I'm bashing Borderlands, because honestly, I think it's pretty awesome - it's just got a few problems.
Loot System:
Destiny > Borderlands
Loot management is much easier in Destiny. You can sell items straight from your inventory, and you're not constantly worrying about running out of bag space and then being forced to go all the way the fuck back to town in order to sort shit out.
Not just that, but in general, Destiny doesn't completely inundate you with loot like Borderlands does. After a certain point, the loot feels completely meaningless in Borderlands because there's SO FUCKING MUCH of it.
In addition, Destiny looks like it has a solid end-game progression path with excellent, hand-tailored gear to barter for - something that Borderlands lacks entirely.
Skill/Leveling System:
Destiny > Borderlands
Again, Destiny trounces Borderlands here. Borderlands takes excruciatingly long to really get going on the class front, but I already have tons of interesting things to do with my classes in Destiny at level 8. A lot of the skills in Borderlands lack impact, and outside of the Assassin, I never really felt like I was getting any kind of huge upgrade when I placed another skill point into my character. Destiny doesn't do the "multiple levels of the same skill" shit that Borderlands does, and as a result, each time you place a skill point it feels like it actually matters.
Leveling is maybe a little too fast in Destiny, however. I'm curious to see how long it will take to go from 1 to 20 in the final game, because as it stands, it feels like I'd be able to do it in a single sitting - I'm not sure how I feel about that.
Art/Visual Design/Graphics:
Destiny > Borderlands
Borderlands art style, while endearing, is quite ugly in certain spots. Visually, on a technical level, it doesn't even begin to hold a candle to what Destiny is doing. I still find myself just stopping and looking around in Destiny, even after putting in over 30 hours during the course of the Alpha and Beta thus far.
The lighting engine, in particular, is absolutely incredible. Borderlands also has dynamic time of day, but it looks nowhere near as nice as it does in Destiny. I think that environmental shadows don't even budge an inch in Borderlands, but I could be misremembering. In Destiny, they smoothly walk across the environments, and it's an amazingly pleasing effect.
I've always been a sucker for Bungie's art styles, and this game is no exception. Even the low-level armor looks great, and is very cohesive as a whole. Some might call it generic, I like to call it consistent. In many MMOs mixing and matching various low-level armor pieces can turn into an absolute disaster (from a visual standpoint), but in Destiny, the armor seems like a natural progression from lower to higher level gear, getting slowly more intricate over time. I'm very impressed, honestly.
One thing I'll give Borderlands is monster variety - in Destiny, the enemies you fight are generally the same size. There's very little difference in the stature of enemies from race to race (at least for what we've seen from revealed media - the Fallen are a little taller than most Hive, the Vex seem to be somewhere in between the two - the Cabal are wider, but still about the same height). In Borderlands, you fight tiny things, and you fight ENORMOUS things. I realize it's unfair to judge based just off the beta on this one, but it seems like it's going to be the case for the final game as well. I'd like to see some more variety among trash mobs.
Gun/Gameplay:
Destiny > Borderlands
The most important category here. Borderlands is floaty and even the most insane end-game weapons just don't have the same kind of impact as even the very first starting rifle you get in Destiny. There's hardly any feedback - gunplay feels almost completely hollow.
Bungie has always been good at gunplay, and Destiny is no exception. Weapons feel weighty, and the combat is incredibly satisfying. My roommates and I are able to just grind for hours in the same spot - something that I HATED doing in Borderlands - because the combat feedback loop is just so fucking satisfying. Critical hits feel like critical hits, and the awesome animation you get for head-shotting enemies is just too good.
I think that the combat is by far the weakest part of Borderlands. It's almost completely thoughtless, especially when you get better weapons later on. Hold down trigger -> sweep the room -> you win. Destiny makes you think about encounters, something that I only ever had to do on maybe one or two bosses in the entire Borderlands series.
e: sorry for the double