The most common answer I see is something along the lines of "it's the equivalent of liking a post on twitter". It seems that this is not the case, as the Mastodon devs seem rather adamant that they don't want "likes" in Mastodon. Perhaps it's a method of saving posts? Well, that doesn't make sense either, since there is already the ability to "Bookmark" a post to save it.
It really just seems like a "Favorite" is just a bookmark that tells the poster, and the public that you bookmarked the post. And even if this was the reasoning -- which is baffling enough as it is -- it wouldn't make sense since the whole point of boosting something is to tell the public that you like a post.
It really seems like the "Favorite" button has no actual unique purpose. In my honest opinion, Mastodon should just federate "Likes" like normal, and be done with it.
The pupose of a Like button is to express that you like the post.
What is baffling about it? The function fulfills exactly the purpose that its name promises.
With a boost, you pass the post to your followers. This is a different feature accordingly.
Like before Boost often seems useful, but hardly in case of bad news.