They're often transcribed as a high back unrounded vowel, IPA ɯ, X-SAMPA M, but they differ slightly. Its kinda like how French and English both have a sound transcribed as [ u ], but they don't sound exactly the same.
Japanese [ɯ]/う isn't technically a high back unrounded vowel. Rather its a high back vowel with lip compression, [ɯ̟ᵝ]. Meaning that The lips aren't fully rounded and protruding as in English and other languages with [ u ] (such as Korean's ㅜ), rather the corners of the lips are pressed together but the lips remain flat. Japanese う also has a tendency to be slightly more relaxed (fronted and lowered) compared cardinal [ɯ]
Korean's [ɯ]/ㅡ is more of a true unrounded vowel. Unlike Japanese's う it remains high, however has a tendency to be fronted.
If you watch the lips and teeth of people singing/speaking japanese, you may notice the lip compression in Japanese and the unrounded-ness in Korean. Furthermore, in Japanese the jaw remains slightly open with the upper and lower teeth slightly apart, whereas when pronouncing Korean's ㅡ the teeth are almost touching.