What is the point of having stars when we can already make our own labels? Aren't stars just another label? It would be interesting to see a use case where using stars is preferable to using labels.
You could consider stars to be another label; you have flexibility in how to use them. Stars are, however, treated a bit differently than labels.
For instance, there is a search operator for stars:
is:starred. For labels the syntax is a bit different:
Stars also have special meaning if you use Priority Inbox. It's one of the signals you can use for the different buckets, such "Important and Starred" or "Starred and Unread".
Stars also have a special hook in the API. I use IFTTT to add an entry to my todo.txt file whenever I "star" a message. This isn't so easy with labels.
Not to mention starred messages can be set to automatically appear on the "Primary" tab (as Vidar points out).
It seems that most people use labels to organize their messages, while they use stars to keep track of messages they want to find quickly, perhaps as a sort of task list.
All that said, if you have no use for stars, there's no reason you have to use them.
Stars act like a "standardized label", that is, a way to mark a message as important, in a way that Gmail understands.
This makes it possible to implement features around starred (=important) messages, such as
If you use inbox categories, messages you mark with a ⭐️ star will also appear in your Primary tab so they are easier to keep track of.
That's one feature that would be difficult to implement with labels.
Based on the answers here I can conclude the following:
When I ask how they are different, the answer is they aren't--really. If labels are a feature, then stars are just a subset of that feature using pre-set setting by google. The most important idiosyncrasies are the API hooks and different search operators. Though I am assuming the gmail API can handle labels in scripts just as well as stars--just differently.
Some other unique points about stars vs labels are a dedicated (read:unremovable) message filter/folder in the left panel and special 'categorization' when using the priority inbox view.
One more thing I noticed playing around with labels is that you can't add labels to single messages in a conversation thread. Stars can be used to mark individual messages.