Yes that is strange. Yes, I see the updates both on the watch face and on the weather widget.
It's a stretch but you may want to try this: Within the settings for the Samsung weather widget, turning on "Use Current Location" if it isn't already.
I am using a Galaxy S6. A strange thing I recall from back when I was using a Galaxy S5 earlier with the Gear Fit was that that setting had to be enabled for the weather update to work properly on the Gear Fit. With the Galaxy S6, it worked properly whether or not that setting was enabled. That seems like a bug to me that maybe was fixed with the S6, maybe exists on the Note 4 and applies to the Gear S2 also.
Also, I have a weather complication on the main face (Modern Utility). I don't know if that makes a difference (it shouldn't, but may be a bug).
I've pretty much tried everything. "Use Current Location" is on in phone weather widget. I guess the only thing would be the phone. Really frustrating. Samsung did fix the low step count with a firmware update so hopefully they are monitoring things and will fix it.