English literary periods are named for monarchs, as with the Elizabethan period, or historical events, as with the Restoration period, or for great literary times or movements, as with the Middle English or Romantic periods. Ben Jonson wrote during the two literary periods named for Queen Elizabeth I and King James I. Both of these periods fell under the broader literary period called the Renaissance period.
About the Elizabethan period, William Shakespeare was the preeminent dramatist during the Elizabethan period and, though he lived and wrote after James I took the throne, is rarely, if ever, associated with the Jacobean period. This is because Shakespeare's greatest renown and the bulk of his work is associated with Elizabeth's reign. Ben Jonson was a contemporary of Shakespeare's and a friend. As such, Jonson too wrote in the Elizabethan period.
However, since Jonson (1) emphasized the moral of his plays, whether drama or comedy, and since (2) his characters were often undeveloped because Jonson preferred shallow typological characters (character who were types of individuals met in life, like the clergyman, the swindler, the innocent young woman etc), Jonson's plays were not popular in Elizabeth's time like Shakespeare's were.
Jonson's greatest work was written and staged after James I took the throne, especially Volpone (1605-06), so even though he began writing during Elizabeth's reign and is therefore an Elizabethan writer, his greatest success was later on during James' reign making him also a Jacobean writer (Jacobean is derived from the Latin version of the name James). Since Shakespeare was the predominant influence in the Elizabethan period and Jonson was not well received then, Jonson had very little (if any) influence during that period.
After James I took the throne and the Jacobean period began, Jonson's fortunes changed. He staged Volpone and found he had become an influence for the younger playwrights coming into their own. Shakespeare was difficult for young playwrights to imitate and model after, but Jonson had such clear-cut rules for structure and purpose, and his Jacobean plays were so well received (unlike his Elizabethan plays), that young writers made him their model for successful play writing.
It is for these reasons that Ben Jonson is classed as both an Elizabethan and a Jacobean playwright. For added clarification, had Jonson been as successful as Shakespeare under Queen Elizabeth's rule, he would, like Shakespeare, be closely associated with only the Elizabethan period. On the other hand, had the preponderance of Shakespeare's work been produced after 1603, which is when James took the throne (1603–25), then Shakespeare would, like Jonson, be associated with both the Elizabethan and Jacobean periods.