According to the Wiki, it's got the same exact stats as the Rocket Launcher, including crits and mini-crits, so you basically have a Rocket Launcher with one less rocket loaded but a +15 health bonus for each target you hit
That one less rocket loaded is huge. In big fights such as the initial midfight, which can often win you the entire round when won, you won't be able to spam the enemy hard enough at the beginning and will lose out on being able to take a positional advantage. Although, it's true that less clip weapons basically have no downside during the middle period of a fight (this is why Liberty Launcher is banned), so they're not completely useless. In fact, if the Black Box was raised to 30-40 restored on hit, it would have a good chance of being banned. Not used more, just straight to being banned.
Also, if the enemy team is smart, they could easily dispose of a Medic+Soldier, heck, i don't even think a Medic+Soldier with Black Box could take down the enemy forces, if your Medic has an Ubercharge ready, that's a different story, but then again, most classes can shred through a varying amount of the enemy forces with an Ubercharge
What do you mean by this?
I don't understand what you mean by less mobility though
With one less rocket, you're not able to rocket jump as much. The fourth rocket in the clip is a very useful tool for engaging and disengaging when the soldier wants to. It's not like you can equip the Gunboats to make up for it either, as now you're losing a ton more damage than you initially were without the Rocket Launcher. The Gunboats themselves are viable (I used them all last season on roamer), but your ammo management is very delicate and is toppled over completely if you lose out on clip size.