(covers information from several alternate timelines)
In 2278, this starship was commanded by Captain Morgan Bateson. In that year, three weeks out of starbase, the Bozeman encountered a temporal distortion in an area of space known as the Typhon Expanse.
The Bozeman encountered the USS Enterprise-D on a collision course. The Enterprise activated a tractor beam to the Bozeman, but it hit the Enterprise's starboard nacelle and its pylons exploded, causing the Enterprise to explode. The explosion, being in such close proximity to the temporal distortion, ruptured the spacetime continuum, trapping both vessels in a temporal causality loop, wherein the Bozeman continually encountered the Enterprise, and the latter ship was destroyed. Eventually, the crew of the Enterprise was able to realize what was happening and avoid the collision, freeing both vessels from the loop. The crew still believed the year was still 2278 when it was in fact 2368, 90 years later. (TNG: "Cause and Effect")
According to the Star Trek Encyclopedia (4th ed., vol. 1, p. 99), the Bozeman was "named for the city of Bozeman, Montana, hometown of 'Cause and Effect' writer Brannon Braga, who also co-wrote "All Good Things...", Star Trek Generations, and Star Trek: First Contact. The registry number was a homage to Steven Spielberg's movie 1941, for which Star Trek model maker Greg Jein provided miniatures."
In the first draft script of VOY: "Flashback", it was stated that Tuvok served aboard the Bozeman. This was changed to the USS Wyoming by the time the final draft of the installment's teleplay was issued.
The novel Ship of the Line stated that the Bozeman was retired in 2371 and became a museum ship permanently docked at Starbase 12. In 2372, Starfleet launched the destroyer USS Bozeman-A (β) constructed and staffed by the crew of the original Bozeman. According to the novel, it was this ship that fought in the First Contact battle. However, Ship of the Line might contradict "Cause and Effect" in some details, such as portraying the Bozeman with an all-male bridge crew (two female officers could be seen on the viewscreen in "Cause and Effect") and claiming that the Typhon Expanse was a regularly patrolled region of space located near the Federation-Klingon border, rather than an uncharted region as in the episode. However, it is also possible that the two female officers seen on the bridge were not members of the primary bridge crew. It is also worth noting that most of Federation space remained "uncharted" (that was, unstudied in scientific detail). Nonetheless, many of these unexplored systems were regularly patrolled as they laid within tactically significant areas of space (e.g., the edge of the Neutral Zone, the Tholian border sectors, etc.).
Another reference to a latter-day USS Bozeman was found in Decipher's downloadable role-playing game adventure "Perdition's Flames" in which the Bozeman was a Saber-class registered as NCC-75032 under the command of Captain Gabriel Bush (β), former Bozeman first officer. No other information was available other than that it was a part of Starfleet Task Force 901 along with an Intrepid-class USS Nelson (β) (NCC-74718) and the USS Sutherland. While the registry of this second Bozeman conflicted with the one mentioned in Ship of the Line, it seemed they were meant to represent the same vessel.