i've only owned/used the BDDI and the VT BASS, both i found to be great pedals, however just on terms of tone, i personally find the VT to be far more versatile and its EQ to be more useful, i always felt that boosting lows on the BDDI only added super lows without being significantly punchy, a by-product of the fixed mid-level perhaps. The BDDI had a nice treble (loved the Presence control) but it just never seemed thick enough in a mix for me . i find the VT to be much fatter, the character control to be infinitely useful as a tone tweaking tool, and the gain/drive to be more pleasing that on the BDDI.
personally i hated the sound of the majority of my effects going into my BDDI (especially my MXR blowtorch), but its usefulness as a DI box meant that i either wanted to compromise on the tone of my FX and have them DI'd or i wanted my DI signal to remain free of other FX and just rely on my amp to convey the Blowtorch, muff etc.
Now, im not saying that i prefer the sound of my effects going into the VT any better, but because it doesnt have the XLR out, i dont feel the need to run it last in my chain, infact i run it first and keep it on all the time at unity volume, so my clean tone is always a beefy dub-svt type tone, and my effected tone is either all fuzz or a blend of VT bass and FX (depending on FX being with clean blend eg: B:Assmaster,Blowtorch, or without eg: Muff, Noisebox)
and now i dont feel like im wasting a DI box just as a Preamp like i did with the BDDI,
VT Bass
PROS
+Mid control
+Character control adds a very useful variable EQ contour to the standard 3-band eq, providing extremes of tones at opposite ends of the spectrum (7 o'clock= sub dub tone, 4 o'clock= clacky overdriven high-end)
+lots of gain available (both volume and drive-wise)
+i find the tones to be more realsitic to what i perceive Ampeg tones are (maybe my limited perception, ive never owned an Ampeg amp) regardless, i find the variations of tone on offer are far superior to the BDDI
CONS
-no XLR out
-no parrallel out
-no clean blend
-not as simple to use (or maybe more so depending on what you're used to)
BDDI
PROS
+Blend
+very good DI box
+ features ie: Ground lift, Line/Level select
+easy to find a quick usable tone
+ tube emulation is really good (but not better than the VT in my opinion)
+Parallel out for BI-amping or for a BDDI-effected DI signal, but a CLEAN amp signal.
CONS
-if you dont like sound of effects into your BDDI, but u still want to DI everything, tough.
(but you could always get another DI, mic your amp or use your amps XLR out, if it has one)
-no mid control- mid is set to a standard freq. (750hz?) and with BASS and TREBLE at 12 it should be flat, boosting BASS and TREBLE will provide a mid scoop
-while many swear by the blend control, i felt that i prefered the 'sparkle' of the sansamp on 100% wet blend, and that otherwise it felt a bit lacklustre
as you can tell many of these PROs and CONs are purely my opinion.
but long story short:
VT BASS