There are advantages for an Air Amb Charity if they share with the police and operate under a police AOC HEMS approval. For a start they get two shifts a day, 7 days a week at a fraction of the cost of even a 5 day single shift dedicated Air Amb unit.It's also a more economic use of the airframe.
As far as I'm aware the only people doing regular night HEMS in UK are the joint units. On the down side they lose the ability to do Air Ambulance work (Inter hospital transfers).
With regard to the number of tasks carried out by Air Ambs when compared to joint units, other than transfers already mentioned I think the difference lies in the way the units are used. Air Ambs seem to often go to jobs that could be dealt with by a conventional ground unit whilst the joint units tend to cherry pick the jobs where the use of a helo will really make a difference.
As an aside, as I understand it when the new Public Services AOC is introduced both police HEMS units and Air Amb units will be working to the same rules and approvals so both will be able to do everything.