In my experience, Console mounts are more sturdy. They are usually mounted to the baseplate of the console. The baseplates are usually secured using both the drivers and passenger seat bolts, not just the passenger side.
The pole is also shorter since it is mounted on the trans hump, since it is shorter, the center of gravity is less.
In my current unit, a 2010 Ford Explorer, it has a console shifter, so I had to use a pole mount. I ended up using a bracket and securing to pole to the side of the console since 85% of my patrol area are dirt roads, so it sways and vibrates.
I am using a Panasonic Toughbook CF-19, with a Gamber Johnson Dock for the CF-19, and also a GJ Mongoose sliding mount, with a GJ Pole and base.
I bought a CF-19 to replace the super heavy Motorola ML900 and its dock that was about 15 pounds alone. The swaying reduced a lot since I went with a lighter computer and dock. About 10 pounds with the CF-19 and its dock, and about 20-25 with the ML900 and its dock.
In all it just depends on preference really. You can buy support arms that attach to the pole and to the floor to add a little more stability. But I would prefer a console mount if I could use it. It would be centered in the vehicle, and closer to the driver, and it would not effect if a passenger sits in the passenger seat. Others usually limit the space on the passenger side.