Kano created a hybrid martial art from existing jujutsu in Japan and named his hybrid judo. What made judo novel was not the introduction of new techniques, but the manner of training them. The critical elements according to Kano are:
- randori (free practice): This is non-prearranged live practice against resisting opponents, in contrast to kata-style training that was common in jujutsu. Some techniques are prohibited in randori due to injury danger. Judo retains kata training where these techniques still exist, but this is frequently peripheral. There is obviously a tradeoff between the advantages of training live techniques and marginalizing more hazardous ones.
principles, not collections of techniques: For example, all throws are studied in three basic parts:
- kuzushi (off-balancing)
- tsukuri (entry)
- kake (finish)
This is in contrast to learning only a collection of unrelated techniques and trying to optimize each individually. By studying principles, it is easier to learn and organize. The principles point may be more important historically than today.
In Japan, it is my understanding that Judo has largely displaced jujutsu, as Kano states in Kodokan Judo, published by Kodansha International Ltd., 1986, p. 19:
Eventually judo displaced jujutsu in Japan, and no one speaks of jujutsu as a contemporary art in Japan, although the word has survived overseas.
I am ignorant of major evolution in jujutsu since Kano's time (which is actually well before 1986). Jujutsu does not have judo's emphasis on randori, so it has more of the "too dangerous to be practiced" kinds of techniques. Jujutsu has a greater emphasis on striking techniques, which are secondary even within judo's often limited kata training.
Brazilian jiu jitsu (BJJ) (jiu jitsu = jujutsu) is a direct descendant of judo and definitely retains the randori-based training model. The primary emphasis in BJJ is on ground techniques and submissions. Submissions end matches, while everything else just scores points. BJJ retains some joint locks that are now illegal in modern judo, for example on the knee and ankle joints.
Foot sweeps are better developed in judo than in jujutsu. These minor techniques are important to developing continuous attacks through combinations in randori. Modern sport judo emphasizes throwing techniques from the standing position and has generally moved away from groundwork.
I have only studied judo directly, so it's necessary for me to organize the answer around judo. I think judo is essential to understand the relationship between jujustu and BJJ, however, because judo has largely replaced jujutsu and BJJ is descended from judo.