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.
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.
Böhm's most important contribution to North German keyboard music is the chorale partita, a large-scale composition consisting of several variations on a particular chorale melody. He effectively invented the genre, writing several partitas of varying lengths and on diverse tunes. Later composers also took up the genre, most notably Johann Sebastian Bach. Böhm's chorale partitas feature sophisticated figuration in several voices over the harmonic structure of the chorale. His partitas generally have a rustic character and can be successfully performed on either the organ or the harpsichord.