The Zoom H1 has cardioid capsules in an X-Y stereo configuration. That's better for music and ambiance. The VideoMic is a mono hyper-cardioid mic. That's better for subjects in front of the camera, and indoor shooting.

The Zoom H1 has no built-in anti-shock mount. That's a problem. You'll need to build or buy something if you want to mount the Zoom your camera and you are holding the camera in your hands. (on a tripod it makes no difference) The VideoMic has a built-in shock mount. It might not be the greatest shock mount, but it's probably good enough.

The Zoom is an audio recorder, with better recording quality than the GH2. But syncing separately recorded audio with your video recordings is troublesome, at best. It's worth it for a musical performance or if you shoot one long take, but hardly worth the trouble for casual recording. The GH2's onboard audio recording is actually pretty good, and more than good enough for casual recording. Hooking the Zoom's audio output to the GH2's mic input is an option, but you'll have a big jumble of cables with an attenuator in there somewhere.

Personally, I would try to use a camera-mounted mic and the GH2's built-in audio recording for everything except musical performances. For musical performances, I would use the H1 mounted on its own tripod. For an on-camera stereo mic, my pick is the Stereo Videomic or Stereo Videomic Pro. For an on-camera hypercardiod, my pick is the Sennheiser MKE-400. These on-camera mics cost a lot more than the H1, and are a poor value for what you get, but for the convenience of recording audio in-camera and having an integrated shock mount, the cost is worth it. BTW you'll need a fur windmuff for all outdoor recording.

Both the GH2 and the H1 can record from an external lavalier mic. The H1 has better audio recording quality, but lower gain, which is a problem for a lot of professional lavalier mics, which tend to have lower sensitivity. If you have a lavalier, just try it on the GH2 and the H1 and see which one gives you better quality. It's kind of hard to find a decent quality lavalier that has higher sensitivity, so you might have to use a lavalier mic that can be adapted to plug into a quality pre-amp with an XLR connector and 48-V phantom power. XLR mic adapters and pre-amps are easy to find, but bigger and more expensive.

I just ordered a cardiod lavalier mic that is supposed to have higher sensitivity. I'll let you know how it works with my GH2 and H1.

the problem in this comparison is the different recorder. Rhode goes into crappy Canon 5DM3 preamp producing a lot of noise. I bought the Rode stereo video mic and to have some ambiance athmosphere in addition to the boom shotgun mic. I was disapointed about the noise and short before sending it back I did a test with the Roland R-26. Way far better than with Canon!