Panasonic G7 - 4K resolution, Micro Four Thirds (MFT) sensor (2.22x crop factor), the 1080 resolution mode shoots at 60 fps, slightly better dynamic range, 17% lighter weight (better when using stabilization equipment), 16% better battery life, image stabilization for photos. The G7 gives you the SAME exact image quality as the $1,400 GH4, except with a few features that you might not care too much about (not waterproof, stops recording after 30 minutes of video).
Canon EOS M3 - 1080p resolution, APS-C sensor (1.6x crop factor).
The M3 is about $168 less than the G7 on Amazon. You might be able to find some good deals on used cameras if you search around.
The G7 shoots in 4K resolution and the M3 shoots in only 1080p. This makes the image of the G7 much sharper, and is especially important when filming things in the distance. When reviewing a few M3 video tests, a lot of the shots were not that sharp. It's important to keep in mind that editing with the G7's 4K resolution videos will require a better computer that can handle the footage, but you can always convert your 4K videos to 1080p or use proxies to edit the 4K footage. It's not a big deal if your computer can't edit 4K footage efficiently, but would add a small benefit to your editing technique if you could.
The G7 shoots 60 fps when recording in 1080p, while the M3 can only shoot a maximum of 1080 at 24 or 30 fps. The G7 doesn't shoot at 60 fps if recording in 4K mode. So, if you film weddings or need a few shots in slow motion with the G7, switch your settings to record to 1080p.
The G7 has slighter better dynamic range than the M3. When viewing footage from the M3, lights were very blown out (there's a large glow around lights). The G7 isn't that great with dynamic range either, but the glow around lights would be reduced by about 20% when comparing it to the M3. This is important when filming images with dark and light sections in the same video. Usually, more dynamic range helps with filming outside light from windows, a bride's wedding dress, the sky, lamps/street lights, or anything else bright.
Video from the M3 seems slightly worse than the G7 with low-light -- the M3 adds more grain in low light. The G7 doesn't add much grain in low light unless you go above f-2.8/f-4.0. The M3 adds significantly more grain at these f-stops -- about the same, if not more, than my cell phone. If you light your scenes properly, then you shouldn't have any grain in low light with the G7, but will have a slightly-distracting amount with the M3. Without lighting (such as at a wedding), the G7 isn't the best in low-light, but with a good lens it's decent. Even with a decent lens, I wouldn't use the M3 to film in low light at a wedding.
The G7 uses a MFT sensor and the M3 uses an APS-C sensor. The sensor affects several things. First, it determines what type of lenses your camera can use without a lens adapter. The M3 uses EF-M lenses and the G7 uses MFT lenses. Before buying your camera, you should research the quality and price of the lenses that it accepts. The ASP-C sensor is wider than the MFT sensor, which gives your video better depth of field (the background and foreground of the ASP-C sensor will be blurrier, which is better.) The G7 has a crop factor of 2.22 and the M3 has a crop factor of 1.6. This means that the M3 records a wider image than the G7. If you're using a 50mm lens with the G7, the actual number of mm is 111 (50mm lens * 2.22 crop factor = 111mm). If you're using a 50mm lens with the M3, the actual number of mm is 80mm (50mm lens * 1.6 crop factor = 80mm). This means that you'll have to move the G7 back a foot or two in order to record the same image size as the M3. This is also important when purchasing lenses, since you'll have to take into account the crop factor when buying a lens with a certain mm number. Check out
http://www.filmhandbook.com/camera-techniques/focusing-depth-of-field/ for more information about depth of field and your camera's crop factor.
Summary: The main pros of the G7 are a sharper image and less grain in low light. The main pro of the M3 is the sensor size, which gives you a wider image and slightly better depth of field. Even though the M3 gives you better depth of field and gives you a wider image, it's clear that the G7's image is significantly better due to the sharper image and less grain in low light. Without even thinking twice about this, the extra $168 is definitely worth getting the G7 over the M3, plus you get to use 4K to allow you to use creative techniques when editing. Cell phones are almost as good now-a-days as the M3, aside from the M3's medium sensor size.
Let me know if you have any other questions!