What's the difference between Ju-jutsu and Jiu-jitsu? In Sweden we only have Ju-jutsu and I have noticed that in the USA they have both Ju-jutsu and Jiu-jitsu and they look like the same thing.
They are the same thing. It's only a matter of romanization (spelling japanese words using roman letters).
As a reference point, here is how it is pronounced in japanese (found on wikipedia).
As to how it is written, it all comes down to how the names were romanized. The most popular systems used today are probably the Hepburn system, the Nihon-Shiki system or the Kunrei-Shiki system.
Of those, the Nihon-Shiki system is the oldest, created in 1885 by a japanese physicist hoping to get rid of the traditional kanji and kana system in the hopes that it would make exchanges easier between Japan and other countries. This system was developped with japanese-speaking users in mind and as such is probably closer to the japanese pronunciation.
In this system, the proper spelling would be "zyuzyutu".
The Kunrei-Shiki system is an evolution on the Nihon-Shiki made just before WWII to incorporate the phonetic changes that naturally occured in the japanese language since the Nihon-Shiki system was put in place. Since it's still made with native japanese speakers in mind, it can, like it's predecessor, induce non-native speakers to pronounce words in an awkward way. As such, even though the japanese government made it the officially recognized romanization system, most people, including some government agencies, prefers to use the Hepburn system.
In this system, the proper spelling would be "zyuzyutu" as well.
The most globally accepted romanization system is the Hepburn system. This system was made in 1908 with a focus on english and italian pronunciations. As such, non-native speakers using this system will generally pronounce words with more accuracy. This system is the most used by non-native speakers, as well as by the japanese Ministry of Foreign Affairs (for passports) and by the Ministry of Land, Infrastructure and Transport (for road signs). There were some reforms to this system over the years, but I do not believe it affected the spelling of this specific word.
In this system, the proper spelling would be "jujutsu".
It is important to note, however, that prior to these systems, there still existed a romanization system created around 1550 by a japanese catholic. Indeed, portuguese traders spread catholicism to Japan around these times and I guess a romanization system could prove useful when dealing with foreigners.
In this system, the proper spelling would be "iuiutçu" or "jujutçu", depending on the version you used.
As we can see, even in officially recognized systems (like Nihon-Shiki and Hepburn), there exist vast differences in spelling. Historically speaking, some systems have written jujutsu using an "i/y" sound in between the consonant "j/z" and the "u". When confronted with non-native japanese speakers that only learned spoken japanese and no written form, it is highly likely that some accuracy in the pronunciation has been lost when they had to write the name of their martial art in their own language. This could explain the many spelling variants of jujutsu we can encounter.
However, if I had to say that a specific one is the "best", I'd go with "jujutsu" (Hepburn), as it seems to be the closest english analog one could get.
In North America (and likely other places)
Generally speaking, practitioners of Japanese Jujutsu will favour the "Jujutsu" or "Jujitsu" spelling, depending on their lineage. On the other hand, practitioners of Brazilian Jujutsu will generally favour the "Jiu-jitsu" spelling, which is much closer to the old Portuguese system and totally makes sense for Brazilians to use.
If you are referring to 柔術, then we can look at the two kanji.
The first kanji is found in 柔道 -- judo. The second kanji is found in 剣術 -- Kenjutsu. Thus, I would opt for jujutsu as being the logical romanji form of 柔術. The other "spelling" maybe viewed as either incorrect or illogical based on this. Thus, I would translate 柔術 as the soft art which contracts nicely with weapon use -- aka hard.
Note: I am not fluent in Japanese and thus could have got the meaning/translation of the kanji wrong.
Ju-jutsu and jiu-jitsu are different romanizations of the same Japanese word(s) 柔術, similar to how we have both Qur'an and Koran from the Arabic الْقُرْآن.
Jujutsu was historically spelled with hiragana2 3 like so: じうじゆつ
Individually these characters are transliterated:1
じ う じ ゆ つ ji u ji yu tsu
but when occurring together, some of these characters represent different sounds (similar to how ch represents a different sound from c and h)1:
じう じゆ つ jū ju tsu
Due to such ambiguities, and that in the early 20th century there were multiple competing romanization standards for Japanese (some misleading), multiple spelling variants (jiu-jitsu, ju-jitsu etc) emerged. Since these spellings were used during the early spread of the art in Europe and the Americas, they have stuck around with e.g. Ju-Jitsu remaining a common spelling in France, Canada, the United Kingdom, and the United States and Jiu-Jitsu in Germany and Brazil.
1. Using the Hepburn romanization.
2. Japanese syllable based 'alphabet' (syllabary).
3. In modern hiragana (adopted 1946) this is spelled じゅうじゅつ.
In Germany those two refer to different things but that is a special case: Jiu-Jitsu in Germany is usually used for the traditional japanese system and related styles while Ju-Jutsu is used for a system developed in the 1960s for German police forces. So in Germany those two are different but that does only hold for Germany because everywhere else the German Ju-Jutsu is called German Ju-Jutsu.
Otherwise I would say it is a question of transcription from Japanese.
I believe these are spelling variations - wikipedia lists both of these as acceptable transliteration of the Japanese word. Of course I'm not fluent in Japanese, so I would defer to someone who is.