WARNING: I, Mario, couldn't cover everything.
Let's-a-go!
Introduction
If you haven't seen Super Mario Run yet, I suggest you watch this video by GameXplain:
World Tour
Just like every other 2D Mario game.
Toad Rally
Versus another player. Your score: Coins + Toads.
Toads can be collected by doing fancy moves. If you win, you earn Toads; and if you lose, then you won't just lose the match, you'll also lose some Toads. Toads can be used in Kingdom Builder.
You can play against your friends, but you have to send them a twelve-digit code. Plus, coins and Toads will not be lost, and if you win, you won't get Toads and you won't be given coins.
Remix 10
You'll play through 10 very short courses. Clearing Area 30 unlocks Princess Daisy, who can double-jump.
Kingdom Builder
Toads from Toad Rally unlock items which you can buy with coins you collect. What's funny about the game is that it reminds me of New Super Mario Bros 2. That game had so much emphasis on collecting coins, and Mario Run also has it. Sure, it's just to buy stuff for the kingdom, but you actually have to buy their houses to unlock Luigi, Toadette, Yoshi, and Yoshi recolors.
More info + My Final Thoughts
The music and art style are the same ones from New Super Mario Bros U. The songs have been remixed very slightly, the biggest difference being the "bah" sounds have been removed. I also love how all characters besides me have their own abilities. Luigi jumps high, Peach falls slowly, Toad and Toadette run fast, and Yoshi can flutter-jump. Oh, and Daisy can double-jump.
Note: Toadette runs a bit slower than Toad.
Conclusion
So, is Super Mario Run worth $10? Yes.
Just as a disclaimer, the game has some things that need to be fixed. For example, you have to be online to play the game. I at least think we should be able to play World Tour, Kingdom Builder, and Remix 10 offline, but besides that, it's a fun game.