Artist's concept of New Horizons as it reaches Pluto.
Here are the relevant facts.
Hubble's angular resolution -- the width of one Wide Field Camera 3 pixel, projected onto the sky -- is 0.04 arcseconds, or 0.19 microradians. (I discussed what this means and why I might want to use microradians in last week's post.)
Pluto appears fewer than 3 Hubble WFC3 pixels across.
New Horizons' highest-resolution camera, LORRI, has an angular resolution of about 5 microradians.
So New Horizons' vision is is about 25 times less sharp than Hubble's. Therefore, New Horizons will have to be about 25 times closer to Pluto than Earth is before its images will begin to improve on Hubble's. Right now it's "only" 4 times closer to Pluto than Earth is. It still has a long way to go!
Pluto will appear 3 pixels across to LORRI when New Horizons is about 160 million kilometers away. How did I get that number? Pluto is 2400 kilometers across, so 3 pixels means each pixel is 800 kilometers. Divide that by 5 microradians -- which is the same as dividing it by 5 and then tacking the word "million" on the end -- and you get 160 million kilometers.
New Horizons is currently a thousand million kilometers from Pluto. When will it have closed to 160 million? The simplest way to answer this question is to use the Solar System Simulator. Set it to view Pluto as seen from New Horizons, then play with dates until you make the distance to Pluto go to 160 million kilometers. (Note that you don't have to keep returning to the form to change dates -- you can change them directly, in the URL.) It looks like it'll be almost exactly two more years for New Horizons to get close enough to Pluto for the mini-planet to appear three pixels across to LORRI. Hubble's view actually spans fewer than three pixels, so New Horizons' view will actually be better than Hubble's a bit earlier than that. A good round date to use is January 1, 2015. So it makes perfect sense that January 1, 2015 is when New Horizons' approach science observations begin!
New Horizons Science Overview
New Horizons will spend most of 2015 training its package of seven imagers, spectrometers, and in situ plasma instruments on the Pluto system.