The difference between the B16B and the B16A is a couple thousand dollars.
And the B16B makes 25 more HP.
Personally, I don't think its worth it.. Especially since you can pick up older B16As with the cable tranny (especially good if you get one with an LSD) and non-OBD ECU (all better to swap into a 4th gen) for a lot cheaper than even a newer B16A, let alone a B16B, wich if you ask me, has a jacked up price because so many people gotta have the "Type-R".
If I could get it for a good price I probably would pick one up, but I wouldn't spend much more than a GSR, and I wouldn't be putting it into a 4th gen. If you ask me, the absolute best B series motor you can put in a 4th gen is a 1st gen B16A. It makes good horsepower, and its cheap. Its a lot cheaper than most any other B series motor/tranny/ECU and the possiblities for upgrades for it are endless.
The problem, and the beauty of the B16B is that its built so well from the factory. It comes with an LSD. It comes with high compression pistons and good cams that make power really high. It comes with a bigger intake manifold. And it should come with a factory header.. Not a heavy cast exhaust manifold like most others. Its a nice motor, but if you want to tune it and don't buy top quality parts for it you're likley going to see a decline in HP rather than gains..
Just what I think.
To your last question: If you got the B16B and put a Type-R sticker on the back (or anywhere) of your car, that would tell me that you're wanting to do a swap for all the wrong reasons.