Actually there will be a slight difference in the height. The amount will depend upon how much the rim width differs from the recommended and how tall the tires are to start with. For the tire sizes we use, it would amount to a few hundredth of an inch.
Also, if the wheel is too wide, you will have excessive wear on the outside edges of the tread. If the wheel is too narrow, you will have excessive wear in the center of the tread.
If you are talking about 315s or 305s, either wheel should be fine. If you're talking about 285s, the 10" wide wheel will be too wide by a slight amount, but will probably be OK. 265s and 245s would too narrow of a tire for 10" wheels, but fine on the 8" wheels.