Times I don’t want to take my phone with me:
When my phone is charging in one room, and I want to walk out of Bluetooth range. Bathroom at work. Yard work outside (if you have a house), etc.
Into any sort of situation where the phone could be in danger. Anything like water, falling and breaking, heat, etc. Going to take a citi bike downtown to run an errand. Try to find a way to carry the phone safely while biking without it falling out of your pocket and breaking, or just leave it at the office?
Any situation where you are in public, and will be forced to step away from the phone. Going to the beach, and don’t want to leave the phone on the sand where it could be stolen. Going to the public pool and don’t want to leave your phone in a locker that will be broken into. Just leave the phone at home since your watch has LTE.
Your phone is occupied as it is being used for some dedicated application. Maybe you are making a video or doing time lapse photography. Maybe you are using it to DJ a party (see link). You don’t want to be cut off from messages, notifications, just because you are using your phone for not-phone.
What if you do bring your phone with you, and it does break for some reason? Gonna have a hard time calling for help, unless your watch is a phone!