As someone who upgraded from the Gear S3 to the Galaxy Watch, I would say the main two improvements are:
1.) Far better battery life, especially since the Gear S3 upgraded Tizen and the OS began killing the battery
2.) Water resistant for up to 50 meters in chlorinated water and salt water
3.) Sleep tracker can now record the four stages of sleep: awake, REM, light, and deep. It also is more robust, in that it doesn't stop tracking your sleep if you toss and turn or get up for two minutes in the middle of the night.
4.) If you got the Frontier (presumably because this was the only one T-Mobile offered initially), then the Galaxy Watch looks far more "classic" and appropriate for business environments
5.) The watch tracks more workouts, and now auto detects rowing machines and ellipticals. I could be wrong here, but I think it only auto detected running and biking before.
Number 4 is mostly aesthetics and personal taste, but 1 is a huge deal for me. Towards the end, my Gear S3 would drop ten percent in just a couple of hours, and when I went jogging and used GPS+LTE, the battery would go from 100% to 60% in just twenty minutes. This made longer runs untenable. Now, the same time spent running only drains 8% of the battery. 12% if I also stream music via Spotify (something I dared not do with my Gear S3). Number 2 is not so big a deal to me since I'm not much of a swimmer, but it is convenient to not have to remember to take off the watch before jumping in the shower. Also if I do get back into swimming at any point, I can now track that in Samsung Health. As for number 3, I've gotten better insights into my sleep habits and have actually started figuring out what I should and shouldn't do to get a restful night's sleep.
I would say that that over all, the Galaxy Watch is only better than the Gear S3 if you're using it for its fitness features or if you use it in LTE mode a lot. I like the fitness stuff, and I have to say it's actually encouraged me to work out more often. But if you just use your watch in Bluetooth-paired mode to get notifications and maybe use the occasional app, then it's best to just stick with it instead of upgrading to the Galaxy Watch. For what it's worth, I don't regret upgrading for an instant.