We have both. After the many recommendations for VB AIOs for newborns, I stocked up on them for my newborn stash, and was NOT disappointed! Perfect fit, very trim, trim through the legs, look adorable, etc. I happen to love the giraffe tag.
I have just one Bum-Ware that I bought because they are reputed to be similar and I wanted to give them a try. My babe is just 3 months old, so I guess that's the infant size that I have? I don't remember.
So, similarities: very trim, comes in solid PUL which is remarkably cute, true AIO, velcro/aplix tabs, easy, absorbent and reliable. Differences: VBs have overlapping tabs; VBs are birdseye, BW are flannel; BW are wider through the crotch than my XS and small VBs; VB has long foldable doubler, BW has hourglass shaped doubler; different sizing (as SEEPAE outlined); because BW are 2-size, I don't get that fit that looks "perfect" - but, each size will last much longer; BW are somewhat cheaper than VBs.
No point in not trying them both!