Main differences IIRC:
Offense:
fWAR uses wRC+ instead of OPS+ for bWAR, both highly correlated and tend to not differ by much, but 1 point of OBP > 1 point of SLG, so wRC+ is a better measure since it accounts for this
Defense:
bWAR uses DRS, fWAR uses OAA (I think, it used to be UZR) and incorporates catcher framing into their defense, bWAR does not.
Pitching:
bWAR uses runs allowed and tries to account for the defense behind the pitcher. fWAR uses FIP + infield fly balls (essentially treating them as strikeouts).