Born in Vilnius, Schneider enrolled in the conservatory there at age 10, transferring to the Frankfurt Hochschule für Musik six years later. His violin teachers included Adolf Rebner in Frankfurt and, later, Carl Flesch in Berlin. Schneider became concertmaster of the Frankfurt Museum Orchestra while still a teenager, serving there from 1925 to 1933. In 1932 he joined his cellist-brother Mischa in the Budapest Quartet as second violinist. He and the group settled in the United States in 1938, but Schneider parted company with the ensemble in 1944. During the next 10 years, he performed in the Albeneri Trio and New York (Piano) Quartet, played duos with Ralph Kirkpatrick and Eugene Istomin, and led his own chamber orchestra. In 1945 he received the Elizabeth Sprague Coolidge Medal for his chamber music work.
In 1950 he brought Pablo Casals out of retirement for commemorations of the 200th anniversary of Bach's death; the result was a music festival in Prades. Schneider also worked with Casals at festivals in Perpignan, Puerto Rico, Israel, and Marlboro, VT, in the United States. Chamber music was the chief focus of all these efforts.
Schneider formed his own string quartet in 1952, performing and recording most of the Haydn quartets, but in 1955 he was lured back to the Budapest Quartet, where he remained until the group disbanded in 1967. During this period and thereafter, Schneider worked extensively as a teacher, and began conducting more actively. He formed yet another ensemble, the Brandenburg Players, in 1972.