Inner tubes are usually good for a range of tire sizes, so you should be fine without replacing the tube as long as you can mount the tire without having wrinkles in the tube.
Mounting a 1.95" tire in place of a 2.00" tire should present no other issues either. Any differences are going to be more likely related to the specific tire/tread pattern more than the size. Perhaps a little more or less traction etc. But no big issue concerning the size.