Dual card slots
Sony made the grip just a tiny bit bigger to accommodate two SD card slots -- one a UHS-II, which has a maximum theoretical transfer rate of 300MB per second, and which you need for shooting full resolution at 10fps -- and the high-capacity NP-FZ100 battery which gives it good life.
One of the things the A7R III did not inherit from the A9 is the drive mode dial which sits on the left shoulder of that camera.
This hasn't changed from the A7R II with one exception; there's now an S&Q mode on the dial for shooting at high or low frame rates for slow or quick motion video. Because the sensor isn't the faster stacked Exmor RS model that's in other models like the A9 or RX100 V, the A7R III's maximum frame rate is 120fps.
The A7R III uses the same viewfinder as the A9 but with the same refresh capabilities as the A6500. That means when you're shooting 10fps it doesn't keep up and you're essentially looking at the last frame shot.
It gets the same joystick controller as the A9 and an AF-on button, plus the recording button has been moved to where it's easier to feel and more accessible with your thumb.
USB and more
The camera has two USB connectors, one USB-C (for charging or faster download) and one older Micro-USB, as well as a PC sync connector. Those are in addition to the rest of the connectors carried over from the A7R II, which include micro HDMI, headphone and microphone jacks.
There are some more minor tweaks to the design of the back. For instance, the programmable C3 button which used to be on this side is now to the left of the menu button.
The full-resolution versions are best viewed in the Adobe RGB color space, and the quality of the shots in the photo gallery is not representative of anything.
Detail, continuous shooting
This is cropped and rotated from the original because of my usual horizon problem and also because someone's arm was in the shot. The full-res version is its ugly unretouched self. I included this because it was shot with strobes which allows you to see the detail and focus available with continuous shooting in its best possible light (look at the hair).