Strict transliteration schemes, like Nihon-shiki, are very literal and systematic (and so VERY easy to teach and learn). They are also lossless in conversion. (You get the exact original JPN when converting from the romaji to JPN.) Phonetic-based system have loss in some cases. (i.e. is romaji "zu" = ず or づ? "ji" = じ or ぢ?) This is why you tend to see cases like "tu" instead of "tsu". Both are correct, depending on what system you've been taught to use.

Westerners tend to learn Hepburn (or some variation of it). I do believe JPN people learn the Nihon-shiki system. Nihon-shiki isn't phonetically accurate, but it is more precise from a technical point of view (due to its lossless nature.) On the flip side, you can guess the ethnicity of a an artist based on their romanization habits ^_~