The big difference between watching and starring a project comes down to notifications.
If you're watching a repository, you'll receive notifications for all discussions: project issues, pull requests, comments on commits and any other comments. If you’re not watching a repo, you’ll just receive notifications for the discussions in which you participate.
On the other hand, if you star repositories, you're showing your appreciation as well as keeping track of repositories that you find interesting without overwhelming your feed with activity notifications.
As stated in this GitHub blog post,
A quick note: activity from starred repositories will not show up in your dashboard feed.
Another thing worth noting is that any repositories you were previously watching can now be found on your stars page. If you want to go back to watching them, you’ll need to change them over yourself. There’s also a new auto-watch feature: If you’re given push access to a repository, GitHub automatically adds it to your watch list.
You can see a list of your starred repos at github.com/stars.
And trending (popular) starred repos at github.com/trending