The two terms are certainly different in meaning, as Mark Beadles and Bob have pointed out before me. However, it is possible to create negative adjectives using different prefixes and obtaining different forms and different meanings. I can think of:
misused [describing something that is used in an incorrect or inappropriate manner] and unused [with the meaning of not used or not accustomed]
maltreated [describing someone/something which is treated badly] and untreated [which, according to the OALD, means 1 not receiving medical treatment, 2 (of substances)
not made safe by chemical or other treatment, and 3 (of wood) not treated with substances to preserve it]
mismatched [said of things or people who do not go together well or are not suitable for each other] and unmatched [which the OALD defines as better than all others]
The list is certainly not complete, but I hope I've been able to clarify the matter a bit. P.S. I don't know if there is a specific term to indicate the two kinds of negations, I think it's just that different prefixes can be used to create a negative adjective.
Following @MarkBeadle's wise commentary, I'll add a few examples of correctly prefixed negative adjectives with different meanings.
dissolved [of a solid material becoming part of a liquid, or of a marriage which has come to an end] and unsolved [which has not been solved, as in a mystery]
disqualified [referring to someone who has been prevented from doing something because they have broken a rule or are not suitable] and unqualified [referring to someone who does not have the right knowledge, experience or qualifications to do something]
dissatisfied [OALD not happy or satisfied with somebody/something] and unsatisfied [OALD 1 (of a need, demand, etc.) not dealt with and 2 (of a person) not having got what you hoped; not having had enough of something]
disinterested [not influenced by personal feelings, or by the chance of getting some advantage for yourself] and uninterested [not interested; not wanting to know about somebody/something]
I wonder if it is necessary for me to remove the initial examples (which however show different meanings for adjectives which are "perceived" as having a negative connotation) in order to reverse the downvote...