To put it very simply:
- Russian Ballet: developed by Agrippina Vaganova as we know it today but originally characteristic of Imperial Russia, its school of reference is the Vaganova Academy. The Vaganova method of Russian ballet is characterized by clean lines, incredibly soft and expressive arms and port de bras, extreme flexibility in the arms, legs and back, and endurance. It is focused on a gradual technical procession: you move to the more advanced steps only after a meticulous learning of the basic ones. The goal is to create clean lines and movements without being stiff but gentle and harmonious
- Italian Ballet: developed by Enrico Cecchetti, its school of reference is Teatro alla Scala in Milan. Cecchetti’s dancers are virtuose and athletic. The goal is for them to learn by internalizing rather than imitating the movements, so barre and exercises are different every day of the week and rigidly scheduled. Cecchetti’s method focuses on balance, elevation, elasticity, musicality, fast footwork, and clean lines
- French Ballet: Defined by Rudolf Nureyev in the 1980s, French ballet is not really standardized but the Paris Opera Ballet is usually considered the school of reference. French ballet is romantic and lyric, fluid, clean, elegant and characterized by Nureyev’s attention to musicality and precision by dancers. Due to this technique’s focus on speed and quantity of steps, the music is usually played more slowly
Try a quick google search for more details and maybe watch Balanchine’s Jewels for a comparison between French (Emeralds, Act I) and Russian (Diamonds, Act III) ballet.
Hope this helps and good luck with your musical!