Here is an interesting thread on Tango's site discussing the two defensive systems:
I personally tend to use fangraphs, but that's more a function of their interface (customizable player page layouts & leaderboards) than a strong affinity for the logic behind their metric. But they do come from different places:
Fangraphs evolved from an analytical viewpoint of understanding the numbers behind the numbers. It's tried to bridge the gap from the "analytics" side of the ravine.
Baseball Reference evolved from a historical viewpoint of documentation the game's outcomes throughout history. As it has become more of an analytics warehouse, it bridges the gap from the "historical" side of the ravine.
Ultimately, I tend to err on the Fangraphs side of the "let's stick to what we can objectively, quantitatively account for" rather than B-Ref's "let's start with what the sum of what happened on the field make adjustments" approach. In large part, that's because the things that we historically have measured and treated as real measures of what happened don't actually measure what we care about (e.g. RBI is only a proxy of a given hitter's performance at the plate and ERA is only a proxy of a given pitcher's performance on the mount; as an outcome, both are strongly influenced by other members of the team. I'd rather start from what I know the individual player did and try to make it as rich as I can than start from a a team outcome and try to subtract out the overlaps. But given that the "truth" almost always lies in between the two, I think there are defensible reasons to prefer either.
That all said, at the end of the day, the cases where the two vary significantly are relatively few and generally not sufficiently different to result in meaningful differences in interpretation. And where those cases do exist, we can probably get to the bottom of it through closer examination and adjust our ultimate conclusion accordingly. In that context, give me the tool that's easier to use, which I think is Fangraphs.