To satisfy means to create or cause happiness or pleasure. It may also speak of fulfilling requirements or meeting obligations. It may be used with or without an object.
The noun form is satisfaction and an adjective form is satisfactory. The prefixes un- and dis- may be added to the noun, adjective, and all conjugations of the verb form.
To be dissatisfied is to have a lack of contentment or happiness in regards to something.
Unsatisfied, on the other hand, simply means to not be satisfied, as in the meaning of being unfulfilled or having unmet obligations.
A person may feel dissatisfied or unsatisfied, but a contract may only be unsatisfied.