Schneider was named after Bernd Rosemeyer, winner of the 1936 European Drivers' Championship. He showed early aptitude for racing, winning the 1980 German Kart Championship, the 1982 European Kart Championship with the national team, and the 1983 African Kart Championship. After progressing through various Formula Ford series in Germany and Europe, he moved into German Formula Three in 1986 and won the title in 1987, also finishing third at that year's Macau Grand Prix.
His Formula Three success attracted the attention of Erich Zakowski, who signed Schneider to drive for the Zakspeed Formula One team in 1988 and 1989. The small German outfit made little impression on the Formula One grid: Schneider qualified for only nine of the thirty-two races he entered — seven of sixteen with the turbo-powered car in 1988, and twice from sixteen pre-qualification attempts in 1989. He also briefly drove for Arrows before departing single-seaters entirely. In the early 1990s he raced Porsche sportscars for Kremer Racing and Joest Racing in the World Sportscar Championship, the Interserie, and at the 24 Hours of Le Mans.
Schneider joined AMG-Mercedes in the DTM in 1992, finishing third in his debut season. He claimed his first DTM championship in 1995 in an AMG C-Class. During the years when the DTM was on hiatus, he drove Mercedes CLK GTR works cars in the FIA GT Championship, winning the 1997 title after taking six victories. In 1998, now in the CLK-LM, he and teammate Mark Webber won five races but the title went to Klaus Ludwig and Ricardo Zonta.
When the DTM returned in 2000 with silhouette bodies and V8 engines, Schneider became the series' pre-eminent driver. He took the championship crown in 2000, 2001, and 2003 in an AMG CLK-Class, finishing runner-up in 2002. Teammates in various seasons included former Formula One champion Mika Häkkinen. In 2006 Schneider claimed his fifth and final DTM title, equalling and then exceeding any previous driver's championship total in the series. He announced his retirement from professional racing on 21 October 2008, at the conclusion of that season.
Schneider returned from retirement for selected endurance events. In 2013 he won four major endurance races — the 24 Hours of Dubai, the Bathurst 12 Hour, the 24 Hours Nürburgring, and the Spa 24 Hours — all in a Mercedes-Benz SLS AMG. He also competed in the Pro-Am class of the Blancpain Sprint Series.
Schneider lives in Monte Carlo. He has children Lisa-Marie and Luca Maximilian with his ex-wife Nicole, who is the older sister of German football manager Oliver Bierhoff.
As a five-time DTM champion and 1997 FIA GT title holder, Schneider is the most decorated driver in the modern DTM era. His pairing of a Formula Three title in 1987 with the DTM's most championship trophies makes him one of the defining figures of Mercedes-AMG's touring car dominance in German motorsport.