I disagree, Korean is a beautiful language to hear. It's so rhythmic, but soft and smooth, not jagged or piercing.

Like people mentioned, the consonants are a dead give away: Korean syllables often end in consonants (some of which don't appear in Japanese, "ng"). The only consonant sound a Japanese syllable can end in (although it is actually it's own syllable - listen for Japanese singers holding this sound), is "n". Aditionally, listen for vowels. Japanese only has 5 vowels, all pure: a i u e o. Korean has those 5, plus several "lazy," in-between vowels (ae, û, ô, etc...) and therefore several dipthongs that don't appear in Japanese.

Japanese and Korean share a lot of words which come from Chinese. Once you learn how a certain Chinese character is pronounced in Japanese and Korean, it's really kinda fun and easy to pick out differences and similarities in the two languages. For example, "promise" is "yaku-soku" in Japanese and "yak-sok" in Korean. Eternity is "ei-en" [J] and "yông-wôn" [K]. The changes are pretty standard, so for instance, if you come across another Japanese word with an "ei"-Kanji in it, it will probably have "yông" in its Korean form.

There are a few other ways to tell the difference, but I think the consonants are the most obvious.

Some common words to look out for in K-pop songs might be: sarang, na, nô, kûdae, kû, kûnyô, byôr, yaksok, yôngwôn, maûm/mam, wae, ônje, mosûb, ôdum, bit, sungan, et cetera...