As you might know , historically speakng Russian school permutates directly from a so called Weimar school, perpetrating a tradition of the greatest pianist of all times, Franz Liszt. I found this analysis on http://www.traditionalrussianpianoschool.com/history.php
In my sense it summs up well the issue:
"Contrary to what developed in the West and particularly in France, the Russian Piano School has always been open to external contributions. It profoundly marked the history of musical interpretation by bringing everything of excellence to the art of piano: loyalty to the text, a polished sound, a natural finger position and a physiological ease with the keyboard.
Piano teaching in France was impregnated from the start by the rise of nationalism. The first piano pedagogues were harpsichordists inspired by military methods and the ideals of the Freemasons. The teaching of piano playing isolated itself more and more by concentrating on “pearl-like” playing. In spite of the considerable efforts of Jean-Louis Adam, Georges Mathias and Alfred Cortot (founder of the “Ecole Normale de Musique” in 1919), the French Piano School represents a unique case in the world where one plays the piano using only the wrists and an attack of the fingertips (in the spirit of harpsichordists in former times), resulting in the characteristic dryness and poor quality of sonority.
By contrast, the Russian School opened itself up from the beginning to progressive external influences and became a synthesis of the best trends of teaching in Europe. In 1861, Anton Rubinstein founded the St. Petersburg Conservatory and engaged the greatest piano professors of the time - the students of Beethoven, Clementi, Czerny, Chopin and Liszt (John Field, Alexander Villoing, Anton Door, Theodor Leschetizky, Vassili Safonov, Alexandre Siloti, etc.). The Russian Piano School was thus to become a school of sound, of « cantabile » playing and of the freedom that leaves a vast space to the interpretor’s personality. Their passion for music and the far-reaching pedagogical ability of the Slavs made up the rest, and that is why this school produced – and continues to produce – so many excellent pianists!
The successors to this prestigious school were Heinrich Neuhaus (1888-1964), Alexandre Goldenweiser (1875-1961) and Konstantin Igoumnov (1873-1948) in Moscow. Each of the three masters had his own progeny, his family tree: Richter, Guilels, Lupu for Neuhaus; Ashkenazy, Davidovich, Feltsman for Igoumnov; Bashkirov, Berman, Nikolaieva for Goldenweiser. The touch of the pianists descended from this school is exquisitely balanced. It fills the sound with a heightened sense of improvisation and of extraordinary liberty."