There are multiple romanisation systems for Japanese, but the most common one is Hepburn. In Japan, Kunrei-shiki is sometimes used (especially by the government), which is designed with Japanese speakers in mind (vs Hepburn which was designed with English speakers in mind).

It's jujutsu in both, but there are varying ways of representing the long vowel on the first "u" -- either omitting it (jujutsu), using a macron or circumflex (jūjutsu), or repeating the vowel (juujutsu).

Ju-jitsu and jiu-jitsu are not correct in any romanisation system that I know of, I'm not really sure how they came about. Probably historical accident.