To make the test very challenging, I used long lenses on the cameras.
On the E-M5 II, I used the Olympus M.Zuiko 45mm f/1.8 (my review) at f/2.8. As the Olympus camera crops the sensor a bit when recording 1080P footage, this lens corresponds to about 60mm in video mode, or 120mm in equivalent focal length terms, which is a quite long lens.
The Lumix GH4 was fitted with the Lumix X 35-100mm f/2.8 (my review) set to about 60mm, f/2.8. I set the lens to 60mm to match the field of view of the Olympus camera, when recording video. The GH4 does not crop the sensor when recording 1080P footage, thankfully.
I used OIS on the Panasonic Lumix system, and IBIS (In-body image stabilization) on the Olympus system. I only had the IBIS mode (mode 2) initiated on the OM-D, not the IBIS+digital stabilization (mode 1).
Both cameras were connected to a Desmond mini stereo bracket:
The video settings were: 60FPS 1080p, with autofocus enabled, using the medium sized centre AF spot, on both cameras
As you can see, the Olympus OM-D E-M5 mark II is vastly better at stabilizing the footage. I have side by side comparisons of the video output below:
The Olympus OM-D E-M5 Mark II video image stabilization works remarkably well, even with fairly long lenses. Using a wide angle lens gives much better results than what has been shown here.
Perhaps this test was a bit over the top: You wouldn't expect to be able to walk around with a 120mm equivalent lens, and expect the footage to be stabilized. But the Olympus camera does achieve it quite well.
On the other hand, the Lumix GH4 was better at focusing during the video recording, using DFD technology and, probably, a faster imaging processor. If you don't care about the stabilization differences, I would definitively get the Lumix GH4 (my review) for video use.