Böhm was born in 1661 in Hohenkirchen. He received his first music lessons from his father, a schoolmaster and organist who died in 1675. He may also have received lessons from Johann Heinrich Hildebrand, Kantor at Ohrdruf, who was a pupil of Heinrich Bach and Johann Christian Bach.
In 1698 Böhm succeeded Christian Flor as organist of the principal church of Lüneburg, the Church of St. John (Johanniskirche). Soon after Flor died in 1697, Böhm applied for an audition for the post, mentioning that he had no regular employment at the time. He was promptly accepted by the town council, settled in Lüneburg and held the position until his death.
From 1700 to 1702 he must have met and possibly tutored the young Johann Sebastian Bach, who arrived in Lüneburg in 1700 and studied at the Michaelisschule, a school associated with the Church of St. Michael (Michaeliskirche).Practically no direct evidence exists to prove that Bach studied under Böhm, and indeed studying with the organist of the Johanniskirche would have been difficult for a pupil of the Michaelisschule, since the two choirs were not on good terms. Yet this apprenticeship is extremely likely. Carl Philip Emmanuel Bach, writing to Johann Nikolaus Forkel in 1775, claimed his father loved and studied Böhm's music, and a correction in his note shows that his first thought was to say that Böhm was Johann Sebastian's teacher.[
Böhm is mainly known for his compositions for organ and harpsichord (primarily preludes, fugues, and partitas). Many of his works were designed with flexibility of instrument in mind: a particular piece could be played on the organ, the harpsichord, or the clavichord, depending on the situation in which the performer found himself. Böhm's music is notable for its use of the stylus phantasticus, a style of playing based on improvisation.