An appurtenance is real property that automatically conveys with ownership. It can be an improvement such as a shed or garage, a right such as a right of way for someone else, or an interest such as use of common elements in a subdivision. An appurtenance is that which belongs to something else. Examples: a back yard goes with the adjoining house as well as the water storage tank, shed, septic tank and right of way for someone else. None of these are directly connected to the house, but in conveying ownership they will NOT be removed.
Tenements are things that are attached to the land physically or legally. Examples include land, houses, barns and the right to collect rent (legal).
The difference between two terms is that appurtenances deal with what is conveyed with ownership while tenements describe what the real property consists of.