Japanese and Korean are actually not phylogenically related. In fact they’re both generally considered to be language isolates, which means they don’t seem to be related to any other languages. This may not be strictly true and there are proposals that there could be a link through a hypothetical language family called Altaic—they’re usually considered unrelated, though.
The “difficulty” of learning any particular language in relation to another is a bit of a meaningless comparison in my opinion. Ultimately, it appears to be the case that all human languages operate according to a common underlying linguistic mechanism—the seeming differences between particular languages are in fact probably actually just different surface realizations of the underlying mechanism with particular options twiddled in various combinations coupled with an idiosyncratic lexicon (i.e., a relatively unique but ultimately arbitrary vocabulary). The nature of the underlying mechanism is the subject of debate, as well as the list of the particular options available, but it seems pretty clear that all languages are utilizing a common underlying cognitive architecture.
That being said, it would seem to me if you define a relative acquisition difficulty metric as something like
D(x, y) = a*O(x, y) + b*P(x, y) + c*L(y) - d*C(x, y)
O(x, y) = cardinality of the set of differences between the language “settings” of x and y
P(x, y) = cardinality of the set of phonemic distinctions present in y but not x
L(x) = cardinality of the lexicon of x
C(x, y) = cardinality of the set of cognates shared by the lexicons of x and y
where a, b, c, and d are scaling factors, then that might give you some semi-reasonable rough correlation in the form of a “distance” between two languages along various axes. The exact definition of O(x, y) is far from clear, however. And even just having knocked this out I’m realizing that there’s almost an infinite number of other things you could throw in here that you would arguably need to know to claim to have “learned” a language: phonological processes, knowledge of and relative productivity of morphological processes, pragmatics, etc. This is why I’m of the mind that any such measure is of dubious utility, because you would need to take into account a lot of factors—most of which we still don’t fully understand.
There’s probably work done on this somewhere but second language acquisition isn’t my area of expertise.
First of all, Korean and Japanese are very similar. They're in different language families because they are not believed to have been evolutionarily related (more closely than other languages in other families), but they are in fact incredibly similar. This might be partly a result of tons of historical contact between Korea and Japan and/or of both sending scholars to China in ancient times, but the mechanism of their coevolution had to be more involved than that. I don't know enough about it, and I'm not sure if linguists do either. However, the fact remains that the two languages in their current forms are closely related, which is what gurkendoktor was asking about in regards to learning time. While the closed class words and inflectional suffixes don't have related pronunciation, the general structure of the languages are ridiculously similar. Word order is identical in many sentences, both use particles or postpositions to mark the function of nouns, both use topics instead of subjects, both allow you to omit the topic if it can be inferred through context, both have a respect hierarchy built into the grammar, both have tons of pronouns and related categories of family words, both have the adversative passive of Chinese, both are agglutinative in the sense that they allow you to add a noun after a verb phrase to form a relative clause that modifies the noun, both have lots of similarly-pronounced Chinese-derived open class words, both have the rare alveolo-palatal fricatives and affricates in their sound inventories as is found in Mandarin, both make the /h/ consonant a voiceless palatal fricative before [i] or [j], etc. Those are just the similarities I can think of off the top of my head. I don't know much Korean, but I'm sure there are a lot more similarities.
As for pronunciation difficulty, I can tell you that from the standpoint of a native speaker of American English who can also articulate every consonant and vowel in Mandarin Chinese, Japanese phonology is far easier for me than Korean phonology. Japanese really only has one consonant whose place of articulation doesn't exist in English or Mandarin (a voiceless bilabial fricative, which is essentially an /f/ but with two lips instead of the bottom lip against the upper teeth) and five monophthong vowels (their position in the mouth stays the same for the duration of the sound), most of which are pretty standard among languages whose vowels are monophthongs. While Japanese does have pitch accent, so does Korean, and unlike Chinese, pitch almost never affects the meaning of a word in the Japanese spoken in Tokyo (there are a couple exceptions) or in the Korean spoken in Seoul. If you want to sound native, you'll have to spend about the same amount of time in either language learning the pitch contour of each word.
Korean on the other hand has more vowels than Japanese, one of which is now pronounced as a diphthong that is very difficult for most English speakers to pronounce correctly (although it's easy for anyone who knows French and easy enough for anyone who can pronounce /y/). The real issue is that Korean has those tense consonants whose specific manner of articulation doesn't exist in any other language. It's the one part of Korean phonology I do not pronounce correctly. Interestingly, I just came across this on Wikipedia:
'An alternative analysis[2] proposes that the "tensed" series of sounds are (fundamentally) regular voiceless, unaspirated consonants; that the "laxed" sounds are voiced consonants which become devoiced initially; and that the primary distinguishing feature between word-initial "laxed" and "tensed" consonants is that initial laxed sounds cause the following vowel to assume a low-to-high pitch contour – a feature reportedly associated with voiced consonants in many Asian languages – whereas tensed (and also aspirated) consonants are associated with a uniformly high pitch.'
So maybe there is hope for us non-Koreans! But yeah, Korean pronunciation is more difficult for most native speakers of English. Also, the grammar requires a little more memorization than Japanese does since there are more inflectional forms in Korean. On the other hand, the Korean writing system is far simpler than that of Japanese since Koreans don't really use Chinese characters anymore at all. It might also take you a few hours or days longer to learn the Japanese syllabaries than the Korean alphabet. As this thread shows, it's incredibly easy to learn to read and write Hangeul.
The “difficulty” of learning any particular language in relation to another is a bit of a meaningless comparison in my opinion. Ultimately, it appears to be the case that all human languages operate according to a common underlying linguistic mechanism—the seeming differences between particular languages are in fact probably actually just different surface realizations of the underlying mechanism with particular options twiddled in various combinations coupled with an idiosyncratic lexicon (i.e., a relatively unique but ultimately arbitrary vocabulary). The nature of the underlying mechanism is the subject of debate, as well as the list of the particular options available, but it seems pretty clear that all languages are utilizing a common underlying cognitive architecture.
That being said, it would seem to me if you define a relative acquisition difficulty metric as something like
where a, b, c, and d are scaling factors, then that might give you some semi-reasonable rough correlation in the form of a “distance” between two languages along various axes. The exact definition of O(x, y) is far from clear, however. And even just having knocked this out I’m realizing that there’s almost an infinite number of other things you could throw in here that you would arguably need to know to claim to have “learned” a language: phonological processes, knowledge of and relative productivity of morphological processes, pragmatics, etc. This is why I’m of the mind that any such measure is of dubious utility, because you would need to take into account a lot of factors—most of which we still don’t fully understand.There’s probably work done on this somewhere but second language acquisition isn’t my area of expertise.