Perhaps I should consult both, but I always use BR. No cherry-picking. I looked closely at fangraph's FIP formula about a year ago -- posted at the time -- and I just don't like it. Their formula is:
FIP = [13 x HR + 3 x (BB + HBP) - 2 x K]/IP + constant
Note that this is a rate/intensity, with no credit for more IP, so I don't think a good measure of value. One might argue that it measures quality of however many innings you happen to pitch, but I'm just not sold on that either, because:
1) I've never bought into the myth of the 3 true outcomes. We know that hitters produce different exit velocities when they hit a ball and we know that pitchers can influence this, as well as the ratio of GB, LD, pop ups, FB. It is a conceit to say that BABIP is solely a dumb luck plus fielding explanation for all not measured by the 3 true outcomes. For one thing, we know that different hitters have consistently different BABIPs. Is it so hard to imagine that over the years, different pitchers have different average BABIPs caused by the quality of their pitching; 2) FIP is not adjusted for ballpark HR factors.
The BR WAR is a counting stat of sorts and it is correct to say it is heavily based on runs allowed compared to those allowed by a calculated replacement level pitcher. It is adjusted for park, league, and fielding. I just like it better as it is closer to my philosophy of how much a hitter and pitcher can control.