I use both, and a Real ICE too.
The ICD3 is definitely faster than the PicKit 3, and I use the ICD3 99% of the time. The Pickit 3 is handy if I'm away from the lab as it's less stuff to carry.
With the ICD3, if you switch between MPLAB and MPLAB X IDEs, then you have to switch drivers which is a bit of a PITA.
Both support the option of powering of the target from the debugger, but I rarely rely on it, I much prefer to provide my own power. The Real ICE does not provide power to the target.
The ICD3 also supports software breakpoints: on the Pickit 3 you're restricted to hardware breakpoints and with that you frequently end up skidding past the actual breakpoint.
Edit: in case you haven't found it, there's an useful functional comparison chart here:
https://microchip.wikidot.com/tools:debugger-comparisonAlthough I have all the advanced trace options for the Real ICE, I've never used them, they're a bit of hassle setting them up. My Real ICE is also one of the first off the production line, and it has some interesting "features" shall we say to be aware of that I have to work around, it's a bit like a dealing with a difficult child at times! The ICD3 on the other hand is very reliable, as long as you remember to switch drivers if switching between MPLAB and MPLAB X.