I've maxed most of the characters in both Borderlands games and played through all but two of the expansions. I really like Borderlands.
But boy, the combat in Borderlands is pretty terrible on its own, and it only looks more dire when you compare it to an actual shooter. Destiny's combat is very strong - there aren't as many procedurally generated elements to the weapons, so you end up with guns that feel right, since they were actually designed, and AI that feels a lot like Bungie's other efforts, which were world-class. Outside the bosses, as long as you're properly leveled, at no point does it feel like the bog-standard high-level Borderlands loop of holding down the trigger and waiting for a health bar to go down.
When they say Destiny is first and foremost an action game, it seems like they really mean it. The shooting element could easily be divorced from the character progression and still be an excellent experience, which is basically the polar opposite of Borderlands. The character-building is less granular (you're not getting +% accuracy by using 1 of 70 skill points - in fact it doesn't expose most of its stats numerically at all), but more fitting for a shooter. There's also significantly less loot than Borderlands, which ends up resulting in a higher ratio of significant upgrades:trash loot - I've always felt like the billion-guns procedural element was interesting but seriously flawed in Borderlands because you get so much garbage compared to actual, meaningful upgrades. Since the weapons you'll be using most are hand-crafted (going from some of their previous statements), it feels more like a great standalone shooter with RPG elements layered on top, not a pure hybrid where one is 100% dependent on the other (and in the case of Borderlands, neither half is particularly great).
And hey, say what you will about the wizard from the moon line, but the writing in Destiny isn't so desperate to be funny that it spews internet memes verbatim every third line, which always rubbed me the wrong way about Borderlands (2 specifically).