While most Starfleet officers had received training in the use of EM weapons, by 2151 the EM-33 began to be replaced aboard Starfleet vessels with the phase-pistol, Enterprise NX-01 being one of them. (ENT: "Broken Bow")
One of the disadvantages of the EM-33 was the need to compensate for particle drift, making the transition to the next generation, point-and-shoot phased pulse weapons difficult for some Starfleet personnel. However, according to Jonathan Archer the two weapons still handled roughly the same. (ENT: "Sleeping Dogs", "Fight or Flight")
The weapon also remained in use aboard Earth Cargo Authority freighters like the ECS Fortunate, as it was seen in the hands of first officer Matthew Ryan during a fire-fight with Nausicaan pirates. (ENT: "Fortunate Son")
Background information Edit
According to the script of "Broken Bow", the pistol was a plasma pistol and the ammunition it fired were plasma bullets,  as the effect was also identified on screen in ENT: "Shadows of P'Jem". In his joint "Broken Bow" audio commentary with Rick Berman, Brannon Braga stated that the weapon was a pulse gun and that he always thought that they looked better than the Starfleet phase-pistols designed for the series and wished they would have stuck with the pulse guns through the series.
These props were in fact later re-used to represent the phase-pistols and an alien pistol weapon in the second season episode "Canamar". The MACO particle rifle appeared to be a rifle version of this pistol prop based on its appearance, as it incorporated a similar barrel and fired similar energy discharges. The MACO pistols were identified as phase-pistols that fire phase pulses in ENT: "United".
In addition, in "Demons" and "Terra Prime", John Frederick Paxton and Daniel Greaves both used weapons similar to the EM-33 and MACO pistols. However, Greaves's weapon fired red energy bolts when he used it to stun T'Pol, while when both men engaged in a firefight with Archer and his crew later on, their weapons both fired green energy bolts. The weapons were used on Malcolm and Trip, but they were only stunned by them.
While the EM-33 pistols seen in "Broken Bow" were grey with black highlights, the pistols seen on the Fortunate were black with grey highlights. This type of weapon is referred to merely as a "plasma pistol", firing "plasma bolts", in the script of "Fortunate Son". The studio model of this weapon was designed by Craig Binkley and Jim Martin.  According to Binkley, it was roughly based on the Israeli "Desert Eagle" handgun.