What are most of these civilian helicopters in North America being
As listed below, Medical and Police use are quite common in the US. The Coast guard runs a fleet of rescue choppers and divers along with their boats. Many news stations also run helicopters for traffic and other areal coverage. There is a fairly sizable market for Powerline, Pipe-Line and Survey related work from rotocraft. There is even a case for moving precarious items.
There is of course a large market here in the US for private, owner flown/operated helicopters in the US. This is at least in part due to the lower costs of private flying in the states. The US airspace system is largely funded by taxes on avgas as well as other federal funding and does not have the same user fee structure that Europe has implemented. In other words here in the states you need to pay to rent or own the plane and gas it up but the use of the air is free. In Europe you will have the same costs with an added cost to use the airspace. In Europe you can also expect AvGas prices to be a bit higher across the board. Although many helicopters are turbine based and run on Jet-A.
More specifically, in North America what are the uses for helicopters
that aren't common in other parts of the world?
Moving people relatively trivial distances... In many places, general sightseeing. If your bold enough, aerobatics. Although many of these uses are also common elsewhere in the world. However due to the nature of aviation laws here in the US, its a bit easier (and somewhat cheaper) to fly small aircraft.
Does this unique demand come mostly from offshore drilling, police, or
Police and Medical is fairly common around the world. Oil rig transport occurs only in places where off-shore rigs are used/drill (many reside close to the US or Asian shores).
Why don't they use helicopters for those purposes in other countries?
They do, for example the UK has a pretty large helicopter medical effort and there is the WestPac life saver helicopter service in Australia. There are efforts in Greece and many other places in the world for that matter.