yo man,
i have both.
bb can get much more sinister and compresses in a legato-ey way if set that way. i hear more stringy marshall zing in the bb.
the direct drive has a drier feel, which can be good if your amp is inherently thick sounding.
but you can also set the direct drive for a really cool mid gain dumbly vibe with the tone knob pulled.
i bought the DD because it was half the price of the maxon OD820 although i liked the maxon better due to increased headroom, more controlled mids, wider bass, and not having to push the gain as much on the 820 to obtain the same amount of single note looseness, but scored a cheap 820 on ebay so the dd is sidelined for now. they are both tube screamer like, but the maxon is more SRV blues, and the DD sounds more snarly in the mids and voxy to me.
and after using the bb for a while, i returned to my tonebone classic for a more sax like lead tone