Utah included an enormous amount of territory (Nevada, today, is the 7th largest state); this made Utah a prime candidate for division, but there were other factors that played into this.
The first was the fact that the Church of Latter-day Saint (the Mormons) were making noises in the 1850s about wanting to have their own sovereign nation - the State of Deseret. The calls for secession (never a winner with the federal government!) combined with the Mountain Meadows Massacre, culminating on September 11, 1857 in the murder of at least 120 emigrants from Arkansas, This act of domestic terrorism demonstrated to many in Washington, D.C. that the Mormons were not to be rewarded with anything but a short leash.
On top of that, LDS endorsement of polygamy ran counter to public opinion, again making the Mormons an easy target for federal retribution.
In response to these issues and events, federal troops made a move to invade Salt Lake City in 1857 to bring the wayward territory into line. Brigham Young, church leader and governor of Utah Territory, called home the faithful to fight the federal intrusion, resulting in the Mormon retreat from the western Great Basin. The promised insurrection fizzled, but it left what is now Nevada devoid of most of its Mormons, leaving the "gentiles" in the majority of a province that was a long way away from Salt Lake City (which was now under federal command).
Gentiles of the western Great Basin began petitioning Congress for the creation of Nevada territory almost immediately. With the gold and silver strikes in what would become the Comstock Mining District in 1859, Congress was presented with even more justification to adopt the creation of a Nevada territory: the LDS Church tended to discourage precious metal mining because it threatened to cause "rushes" which brought could bring in thousands of non-believers, swamping the Mormon communities. The perceived hostility of the Mormon Church against mining was yet another reason to create Nevada in March 1861 (one of the last acts of President Buchannan was to sign Nevada into existence). Because of explosive population growth, the success of the mining, and Lincoln's perceived need for more electoral votes from a pro-Union state, Nevada became a state in 1864.
Subsequent precious metal strikes along the Nevada-Utah border inspired Congress to take two large slices of land away from Utah, adding them to the young state of Nevada, creating the present boundary between the states.