I believe the main reason is for navigation. They need a stable frequency reference because they use its delta, due to Doppler shifting, as the measure of the spacecraft's radial speed relative to earth.

From the paper, even this amazing stability isn't good enough for their navigation needs, by itself:

    7. NAVIGATION SUPPORT

    The downlink frequency from the New Horizons transceiver is
    derived from the USO, as discussed previously, but even this
    excellent frequency reference is not sufficient to support
    precise and accurate Doppler velocity measurements at the
    target level of 0.1 mm/s throughout the mission. In order to
    remove the impact of frequency drift and bias in the
    reference oscillator, New Horizons has implemented the
    noncoherent Doppler velocity measurement approach [2] which
    was successfully demonstrated in 2002 during the demanding
    Earth orbit phase of the CONTOUR mission [3].