I'm providing a limited view on a very large topic, so please take with a grain of salt. Also note that I work for BCG so there could be inherent bias at play.

Specialization

If you're interested in PE, Due Diligence, etc. Bain is known for this and has many opportunities in this areas. BCG has a better reputation for the rest, but of course it varies by office, Partner, etc. Specialisation occurs much more at the regional level as well (New York = healthcare+Finance, LA = media and entertainment, Dubai = energy, government, tourism).

Reputation

Bain is viewed equal to BCG in the US, but, generally, BCG has much more name recognition and brand awareness internationally. I can say that, in Australia, BCG is well known and Bain is not.

Culture

Of MBB BCG is known to most strongly support work-life balance. But this requires a completely new post. A lot of material already exists on culture - suggest you look through the PrepLounge forums, poets and quants, etc. (Not hard to find at all)

Growth

Can't speak to this.