As it seems, "sponsored" and "promoted" are just two different names for the same thing. I think @trincowski explained it best in his comment.
In other words, while from Esteem App side members would hit the "Promote" button, readers would see the article (from time to time) marked as "sponsored."
So, in short, that's basically also the answer I was looking for - What this "Promote" button would do? Simple answer would be, give the post some more visibility (like if you would promote your twit on Twitter or some other post on FB or elsewhere), at the same time not guaranteeing any success or bigger earning.
Exactly.
Then you have the boost option which will increase your earnings but with that option alone, no one outside your circle will see the post.