I'm having a hard time fathoming the difference between incomprehensible and unintelligible.
After some research, I assume that
incomprehensible would be used to express the inability to understand a problem and finding a solution for it or to fathom a process.
whilst
unintelligible rather emphasizes that a sound or writing is difficult to grasp or understand, for example due to noise or crabbed handwriting.
Is this correct?
The most straightforward way to try to distinguish between these two terms is to check their definitions in one or more good dictionaries. Here are their definitions in The American Heritage Dictionary of the English Language, fifth edition (2011):
incomprehensible adj. 1a. Difficult or impossible to understand or comprehend; unintelligible: incomprehensible jargon. b. Impossible to know or fathom: incomprehensible mysteries. 2. Archaic Having no limits; boundless.
...
unintelligible adj. Difficult or impossible to understand or comprehend; incomprehensible: unintelligible remarks; an unintelligible prose passage.
According to AHDEL, then, there is no difference between the meaning of unintelligible and meaning 1a of incomprehensible; but incomprehensible also has the meaning "unknowable or unfathomable," which unintelligible does not.
Merriam-Webster's Eleventh Collegiate Dictionary (2003) offers this definition for incomprehensible:
incomprehensible adj (14c) 1 archaic : having or subject to no limits 2 : impossible to to comprehend : UNINTELLIGIBLE {incomprehensible instructions}
Regrettably, the Eleventh Collegiate doesn't provide a definition for unintelligible, even though in normal practice its use of all caps in a single-word definition signals that the reader should refer to the entry for the word thus capitalized to find the full definition of that word.
The Eleventh Collegiate does have an entry for intelligible—and presumably its omission of a definition for unintelligible indicates that MW views the definition of unintelligible as boiling down to "not intelligible." Here at any rate is the Eleventh Collegiate's entry for intelligible:
intelligible adj (14c) 1 : apprehensible by the intellect only 2 : capable of being understood or comprehended {jargon intelligible only to the initiated}
Unfortunately, the negative of definition 1 above is a bad match for the actual meaning of unintelligible because it suggests that something that is apprehensible by the senses as well as by the intellect—and therefore not "apprehensible to the intellect only"—might properly be characterized as "not intelligible," and therefore as "unintelligible." This is clearly a false line of reasoning, however, occasioned by the Eleventh Collegiate's sloppy treatment of unintelligible.
What MW really intends to offer as the definition of unintelligible, I think, is "not capable of being understood or comprehended"—that is, the negative of definition 2 above.
Merriam-Webster's full-size Webster's Third New International Dictionary 1986) does a much better job of identifying the different meanings of the two words:
incomprehensible adj 1 archaic : having or subject to no limits : ILLIMITABLE [citation from Richard Hooker omitted] 2 a : impossible to comprehend : lying above or beyond the human mind {the incomprehensible mysteries of creation} b : being beyond the powers of comprehension of a particular mind : UNINTELLIGIBLE {an incomprehensible subject to him} c : being beyond ordinary comprehension : UNFATHOMABLE {incomprehensible moods} {a whimsical incomprehensible person} 3 obs : impossible to catch or hold
...
unintelligible adj : not intelligible : difficult to comprehend : OBSCURE
It thus appears that the Third New International, like AHDEL, recognizes that unintelligible and incomprehensible share a meaning (definition 2b in the Third New International) but that incomprehensible has a much wider range of possible meanings. Especially interesting is the Third New International's observation that unintelligible applies to particular minds, whereas incomprehensible may refer to such minds or to the human mind more generally.
Conclusion
To a considerable extent, the words incomprehensible and unintelligible have overlapping meanings; and a reader who consults Merriam-Webster's Eleventh Collegiate Dictionary (for example) in hopes of reading a lucid explanation of how they differ is likely to be disappointed. But other dictionaries suggest that the main difference between the two words is that, while unintelligible is essentially interchangeable with incomprehensible in the sense of "difficult or impossible to for a particular mind under particular circumstances to understand," incomprehensible may also convey a more abstract or generalized notion that the thing so characterized is beyond human understanding.
Sven's answer evidences the extensive overlap in the meanings of the two words, and that without explicit usage sections which address the difference, dictionaries can do as much or more to tangle rather than untangle the two words.
It may add value then to read about the words' particular historical usages in various academic and professional fields.
For example, 'comprehension' is conceived differently in fields such as logic, language learning, and mathematics, and 'intelligibility' is conceived differently in fields of philosophy, linguistics, and communication.
