The LTE-enabled Apple Watch can do a lot of things without connecting to an iPhone, but it can handle calls and SMS/MMS.
SMS / MMS
According to Apple's document on using Apple Watch without iPhone:
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To receive SMS/MMS messages on your AppleWatch Series 3 (GPS + Cellular), your iPhone must be turned on and connected to Wi-Fi or mobile data, but it must not be near you.
In other words, if your iPhone is turned off, in flight mode or not connected to the Internet, SMS / MMS messages will not appear on your watch, even if they are connected to an LTE network.
The reason is that SMS and MMS do not work like phone calls and iMessages.
CNET editor Scott Stein has made it clear thatyour iPhone may lie on your bedside table 50 miles from you, and you can still text messages on your watch if your phone is connected to the Internet.
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IMessages messages on the watch will always work.
Phone calls
Phone calls on the watch can always be received when connected to LTE.
According to Apple:
If your iPhone and Apple Watch Series 3 (GPS + Cellular) are both connected to a cellular connection, then the watch can perform all its functions, even if the iPhone is not with you.
Tip:Turn on Wi-Fi calling to receive and make phone calls on your watch even when it's off the LTE network and connected to Wi-Fi.
Notifications
If you use a watch without an iPhone, then notifications from third-party applications like Instagram, Twitter, Facebook, etc. will not come to them.
"Apple hasn't said anything about this changing in the future," – writes CNET. To receive notifications from third-party apps, your phone must be nearby because they come through iCloud.
In addition, in order to work at a distance from the phone, applications must support the watchOS 4 API.
For example, the Slack application will not work on the new watch, since it uses older APIs. Apple's own applications like Maps, Mail, Messages, Weather, etc. will always work.