Well, a number quite often used is 2 million however I've seen the number 5 million also used, 19.192 million downloads on sourceforge. Google comes up with 55 million so it really does depend on your source. There is no accurate way of working it out as sometimes people download phpBB and then use it multiple times or not at all, there are multiple places to download it and sometimes people give it to each other, not everyone uses the 'send statistics' feature added in 3.0.6, some boards aren't indexed by google, some boards are on internal servers, some remove the credit line, some are on freehosts, some have multiple boards on one domain, there are some phpBB forks etc. Then if you count boards on local PCs then I have over 100 (all for testing MODs etc.)
There is no accurate way to tell as with all open source software. Propriety Software can give you a number of active licences but that doesn't count nulled installations or local installations.
I would take an educated guess that there are probably more phpBB installations than vB installations as its open source free software but they don't release numbers and we don't know a number so there is no way to compare.
And phpBB is constantly being downloaded more times (for example
sourceforge, which isn't our main downloads location anymore still gets around 400 downloads a week). The internet is always expanding.
Disclaimer: None of the numbers mentioned in this post are necessarily true, as I said, there is simply no way to know.