Bethesda usually writes the Brotherhood as this big army badass faction whereas obsidian sees them more as basement dwelling weirdos who solely exist to hard technology.

Which do you prefer and why?

Tough call. I'd say Bethesda's because in Fallout 3 and 4 (I'm not sure about the other ones because I didn't play much of them) their power is clear and showcased to all of the wasteland. But Obsidian's take is the Brotherhood of Steel as bunker hermits. Bethesda's in general, in my opinion, fits them more, since they are very powerful and technologically advanced, and prove it through their presence in both FO3 and 4.

To sum it up, Bethesda's.

I like the Obsidian portrayal of the BOS its more like a religious order then a military merc group. The Obsidian take is more like a group of silent protectors then a nazi like "wipe them out" philosophy.

Honestly kind of hard to say for me.

FO3 had the brotherhood as protectors of the wasteland with some groups still following the old ideals of the brotherhood on "securing advanced technology" and I liked that.

FONV had them as mostly defeated and paranoid about being attacked but still overall good people. Why they never just left I don't know

FO4 had them acting more as assholes who more or less wanted to keep everyone else a caveman rubbing two sticks together. And then take the sticks away when they make fire. Then shoot them

Bethesda's take. It makes sense that a faction that has existed for hundreds of years, has a large army and has spread its influence all around the US.