"Harm": people OR things can be harmed- physically or mentally damaged:
--- The bank robbers were anxious not to harm anyone.
--- The burning of fossil fuels harms the environment in which we live.
Both "harm" and "damage" can also be used as nouns, in various expressions.
"Damage" is usually physical harm done to an inanimate object, or to something abstract (economy, hearing, reputation etc).
----The car was so badly damaged in the accident that it was barely worth repairing.
---- His reputation as a defence lawyer was damaged by the scandal.
However, one exception is the expression "brain-damaged" which of course refers to a person, not an object.