Man, is that way out in the black. The probe is now closer to the orbit of Uranus than it is to Saturn, though both planets are over a billion kilometers away from New Horizons right now.
The solar system is frakkin' BIG (if I may mix my colorful scifi metaphors). If you're still not sure just how roomy things are out there, even at its current speed of 16.5 km/sec (10 miles/sec) -- fast enough to cross the entire United States in five minutes -- New Horizons won't pass the orbit of Uranus until March 18, 2011, more than a year from now. Neptune's orbit isn't until August 24, 2014.
One thing to notice: from this point of view, planets revolve around the Sun in a counterclockwise fashion. Given the position of Pluto, you can see the two are heading for a close encounter soon. Well, for a sufficiently broad definition of "soon": July 14, 2015.
Space is big.