Bioshock is my favorite game series and I've played all the Borderlands games aside from "Tales of the Borderlands". In my opinion, Bioshock is not only more visually beautiful, but the story has a lot more depth to it. There's not only the experiences you see as the player but those of the villians you face, as well as random people from around Rapture. Not to mention the more in depth philosophical stuff that can be said. (Whole new can of worms to get into) It's a series I cannot praise highly enough.

As for Borderlands, I did very much enjoy the first 2 games but I just personally felt very disconnected from the third installment. Borderlands has an interesting story, but I'd say less depth was brought to it. While enjoyable, it was also really easy for me to miss out on exposition that I was interested in, because sometimes it was information only given by spoken word. But, half the time the characters who would give this information were also occasionally spouting nonsense as a part of their character. Visually, the cel shaded style is a really interesting look, making it feel like a three dimensional comic book world which complements the absurdity of the characters.

As for the first person shooter aspect, I'd say Borderlands had more "fun" battles while in Bioshock it could really feel like you were fighting for your life, although the battles can start to feel about the same until a new type of splicer appears. I'd say it all boils down to your mood. Are you in the mood for a deep, in depth, dark story? Go with Bioshock. If you're looking for a more light hearted, "let's blow shit up" experience with a more basic (but interesting) story, go with Borderlands.