I actually owned the LG G2 and switched to the Z1 compact.
At first I actually wanted to return or send back the Z1c because I had got used to the bright and beautiful display of the LG G2. The Z1c did offer over the G2 the following:
1) Size: I could actually touch all points of the screen using one hand. Fit easily in my pocket with a case on. I'm a big guy, so the G2 fit well too but I never had a case on as the added bulk would have made it uncomfortable -- always bothered me that I didn't have a protective case.
2) The expandable memory: It's just great having my media files on the 128GB micro SD card I have in there.
3) The waterproof feature is awesome: I usually treat my phones well, but I also want it to be functional e.g. using it in the rain and not having to worry
4) This was the clincher for me; the camera: I know there are arguments that the G2 has a great camera, but I believe I had a bad unit and my shots were ALWAYS blurry in low light. I wanted to take fast shots with good quality. Granted the Z1c is not perfect, but it definitely was so much better!
So essentially you have to ask yourself what you want in a phone based on what you use it for. I use mine like a mini computer (hence android choice) and my main camera (I got the QX-10 with case attachment and I got amazing shots at the recent car show). The only thing I say I miss from the G2 is the IR blaster.. just a cool feature when you can't find the remote (or visiting friends and you want to control their appliances).
Ps. I got used to the Z1c screen after doing some white balance adjustments and not looking at the G2 screen anymore