Tone-wise, i'd boil the difference down to this :
The Bass Driver has a pre-gain tone control called 'presence', which provides a real sharpness to the sound - almost working like a harmonic exciter. However, there's no way of doing a pre-gain bass cut, so high drive settings sound very buzzy - which I would call a disadvantage. However, it does have a blend knob. Finally, the Bass driver sound is scooped around 800Hz.
The VT Bass has a pre-gain tone control around 1Khz (the 'character' knob) which also cuts lows. This enables a much wider range of usable high-gain sounds - things get 'clangy' rather than 'buzzy'. It has a mellower tone overall than the bass driver. It has a post gain mid control too, which the BDDI lacks.
Both will be able to give you a somewhat 'scooped', larger-than life, sound with mild tube overdrive. The Bass driver has the edge if you want to take that to extremes or if you like very bright sounds, or you like a tight feel.
The VT bass has a looser feel but is better (IMO) for high levels of drive due to the nature of the character knob, and is also easier to set to mid-heavy sounds.
My personal preference at the moment is the Bass Driver, as I like the 'hoojness' of it... But the VT bass does almost everything the Bass Driver does and more besides. So the VT bass would probably be the 'If in doubt' option.
Disclaimer - my experience is with the bog-standard VT Bass and BDDI, so I may be missing a trick or two that the posh versions can do.