Ballet Dance
Ballet dancing was developed in fifteenth century as performance dance but with the passage of time, with the inclusion of music and choreography in it's production, it became popular as concert dance. Trained dancers in the company of famous composers or professional orchestras perform ballets. Ballet dance studios all over the world teach this dance form after years of training; they would teach this dance to children at the age of 3 and above, for them to become professional ballet dancers with technique.

Jazz Dance
Jazz Dance was popularized as a vernacular dance of African-Americans when Africans were brought to America through slave ships. The traditional Jazz dance was converted into modern Jazz in 1950s with Caribbean dance style in it. Jazz is an energetic fun dance form which included big leaps, quick turns and footwork. The grateful balance in the movement of Jazz dancers enable them to attract viewers attention. A professional dancer has to have high energy and low center of gravity.

Vocabulary and similarities with Jazz and Ballet
Pas De Bourree - A sideways step in which one foot crosses behind or in front of each other.
Pirouettes - A rapid whirling about of the body; a full turn on the toe or ball of one foot.
Releve - To rise.
Grande - A large kick.
Jete - A type of jump from one leap to another.
Saute -  A jump on one foot.

Ballet and Jazz are similar to each other as the body alignment, various postures and cross over steps are similar in both the dance forms

Differences
Ballet had a long history of its developments as classical dance form which took several years to get refined into present form as compared to Jazz dance. The quality of movements, emotional content and telling storylines.

With Jazz, it is considered to have modern dance form even if the classic technique is deep rooted into it. The expressions of its versatile music with the change of generations had it made more contemporary. Jazz is incorporated with more dancing ideal, such as hip-hop and break dance.