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"Disgraceful" and "ungraceful" are both derived from negations of "graceful".
Wiktionary describes disgraceful as
and ungraceful as
Are they two different kinds of negations? Possibly the latter saying that there isn't much grace, while the former is saying that the exact opposite of grace exists? If so, are there terms for the two kinds of negations?
Yes, they are different in how the negations are constructed.
Ungraceful is un-graceful: not graceful.
Disgraceful is disgrace-ful: full of disgrace.
(that is, "disgraceful" is not properly a negation of "graceful")
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.
EDIT
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...
I agree with Mark Beadles that, in your example, the root words are actually graceful and disgrace.
But you also asked about whether dis- and un- are two different kinds of negation. They do have some of the same meanings.
According to the Online Etymology Dictionary, un- is considered either a prefix of negation or a prefix of reversal. Un- means: not; contrary to; opposite of.
Also according to the Online Etymology Dictionary, dis- is a prefix meaning 1. "lack of, not" (e.g. dishonest); 2. "do the opposite of" (e.g. disallow); 3. "apart, away" (e.g. discard). Another dictionary gives these definitions for dis-: Absence of; opposite of: disorientation. Undo; do the opposite of: dislocate. Deprive of; remove: dismember.
The similar meanings of the two prefixes can make it confusing to know which to use to form a negation. In general, un- is more widely and freely used than dis- (per the dictionary definitions I linked to). Dis-, though, does have a meaning of "putting away from" that un- doesn't.
I agree with the many discussions about dis and un pairs. However, in this context, I think the distinction derives from different meanings for the word grace. The Compact Oxford English Dictionary defines grace as
1 smoothness and elegance of movement:
she moved through the water with effortless grace
2 courteous good will:
he had the good grace to apologize to her afterwards
3 . . . the condition or fact of being favoured by someone
he fell from grace with the tabloids after he was sent off for swearing
Ungraceful refers to meaning 1. Disgraceful refers to meaning 2 or 3.
As it has already been said, the root words are not both graceful but rather graceful and disgrace.
Mark Beadles has pointed out that disgraceful is not a proper negation of graceful, and this is generally the case with the two prefixes dis- and un-.
They are generally not used in the same way. The prefix un- is only rarely used to negate nouns or verbs, eg. unrest. However, the prefix dis- is usually prefixed to a noun or verb which can then be used to construct adjectives and adverbs. You can see this by the fact that ungrace is not a word, and dis- cannot be added to adjectives with no noun/verb root, eg. unwell vs diswell.
They are very different kinds of negation, and you should consider what you are actually trying to negate. For example, disqualified has the root disqualify which means to do the opposite of qualify, and so the action is being negated. On the other hand, unqualified has the root qualified, meaning the adjective is being negated, in other words the state of being qualified is negated.
I disagree with existing answers saying OP's pair of words differ because they're constructed from disgrace + ful and un + graceful. Whilst this may be a useful way to understand/remember the different meanings, it's not how the two forms came about.
I call attention to Merriam-Webster's definition ungrace: lack of grace, and several thousand instances of the word in Google Books from which it should be clear the specific type of "grace" that's lacking is the religious state of unearned divine sanction.
It just so happens that "grace" has several meanings (divine grace, elegance, an exemption, thoughtfulness, etc.). In such situations, multiple alternate but superficially equivalent inflexions (dis-, un-, etc.) often end up being associated with one or other of the different meanings.
Another example is disaffected: alienated, disloyal, discontent, and unaffected: unchanged or lacking affectation, sincere. Note that in this case there are still two fundamentally different senses of "affect" covered by the second form. Also disinfected / uninfected, and disarmed / unarmed, etc.
It's worth noting that in some contexts there's a tendency for the dis- prefix to convey an "active" sense of something having been explicitly negated, removed or denied, where the more common un- may simply mean not having, in a more neutral/generic sense (lacking, rather than removed).
Where a word with multiple meanings is used with both "negating" inflections, the above tendency often influences the natural divergence which normally occurs between "synonyms". Thus disgraceful/disaffected/disarmed become associated with being actively denied the state of grace/social inclusion/weapons, leaving ungraceful/unaffected/unarmed free for other senses.
They are very different.
Disgraceful refers to "falling from grace", an image of evil being cast out of heaven, so it means having your reputation damaged, being shamed. Ungraceful simply means, "lacking grace", in a more literal, physical context.
Ungraceful means "merely clumsy." It carries no ethical connotation.
Disgraceful implies moral opprobrium.
