If you google for "B-lupa", you'll find tons of articles about how the B-status residence permits are evil, how they all started with the bad, bad Immigration law of 2004, and how the people who get them don't have any possibility of having a normal life in Finland, and how the whole thing is an affront to human dignity: one cannot work, cannot study, cannot get money from Sossu, etc.
Funny. I've had a B-status residence permit in 1994, and for a number of years after that. They have existed long before the law of 2004. I think I've had them for 8 years or so. I would have liked A-status much better, of course, but all in all my human dignity survived without much damage. I've also studied during that time, and worked full-time, legally.
B-permit just means that the stay in Finland is temporary. They are given to temporary workers, sometimes to permanent workers for a while, to students, to asylum seekers who don't get asylum but cannot be returned, etc.
In my time B-permits were often given to the foreign workers with permanent jobs for the first two years, after which they could switch to an A-permit. The new law was supposed to fix this (and give A-permits to everyone who has a permanent job in Finland), but I don't know whether or not it did.
What you can do with a B-permit depends on the reason for the permit, on the time when you got the first one, and on the city where you live. The people who got a work-related B-permit obviously can work. The permit is usually limited to a particular field, and before 2001 was limited to a particular employer, which was quite a drag. The study-related B-permit gives one the right to hold part-time jobs, and if one wants to have a full-time job one has to apply for a separate work permit.
The extent of social safety net one gets with a B-permit depends on the time and the city. I was told by the Kela people that the only social security I could get were the health services and the apartment subsidy (never understood why the apartment subsidy), but I've heard of people getting the same services as citizen and permanent residence, and of people who were denied even the health services.
The "not-really-refugees-but-can't-deport-them" B-permit does not include a right to work for the first year. I've heard several times that these people can apply for a separate work permit. I don't know how often they get it.
They don't have an official city of residence, so I suppose studying in secondary schools can be a problem. Studying in universities AFAIK does not require any permit except being admitted there, or any city of residence.
In a way, I agree with the people who protest against giving B-permits to the asylum seekers: there is not much sense in having temporary refugees, so one can as well either give those people a permanent permit or find some way to kick them out. There are enough people living on taxpayers' money who don't want to work - why should we also have to support people who are not allowed to work?
Nevertheless I find it disturbing that the anti-B-permit crowd seems to be completely unaware of what the law says, of the existence of all the other (not asylum-related) B-permits, of the existence of B-permits under the previous law, etc.