As others have said, the two share the franchise name and its core concepts, but they are really very different games.
Forza - Italian word meaning “force”. The title Forza Motorsport is a mangled English-Italian hybrid that loosely means “forceful (agressive) racing”
Forza started out as a direct competitor to PlayStation’s exclusive Gran Turismo racing series. Despite what anyone here tells you, Forza is Halo for gearheads. It’s an Xbox-exclusive game meant to move Xbox units by pulling people away from other racing games. In that regard, Motorsport focuses on cars and racing in a somewhat realistic fashion. Over the years, Forza has pushed for more real-world tracks and physics (though the physics are heavily debated), and as you can see from the newest installment, Turn 10 actively works with auto manufacturers to bring real world cars and parts into the game. Throughout the series’ history, it has focused on various aspects of the real automotive industry and racing culture.
Forza Motorsport - “Real-life racing simulation with arcade controls”
Motorsport presents some unique challenges in choosing a car, tuning it, and of course, racing it. You will face cars with vastly better or worse performance than your own. Events will limit certain aspects of your car, such as the drivetrain, engine placement, manufacturer, and many other factors. Events will explore a plethora of racing types: point-to-point, hot laps, circuit races, endurance races, multi-class, track days, etc. In this way, Motorsport focuses on the cars as a tool, and how you use them to achieve victory in a race.
That’s not to say Motorsport is completely devoid of fantasy and fun. Many installments in the series have featured fantasy tracks (FM6 has I believe 3 or 4), as well as “playground games” and other multiplayer modes that would never be done in a real car. However, on the whole, Motorsport is concerned with realistic(ish) races and cars.
Horzion - “Let’s get some cool cars and do whatever we want!”
Horizon is Turn10’s attempt to map the simulation-based aspects of Forza into a fantasy setting. Horizon basically dangles the carrot of Motorsport in front of you, but it removes most of the limitations of Motorsport and drastically shifts the focus of the game. You will never face cars that are vastly different performance from your own. The events limit the type of car, but don’t really limit things like drivetrain, engine placement, etc. There are very few race types in Horizon: Point-to-point, street race, cross country (off-road), and circuit. Horizon is not about cars as a tool, Horizon is about cars as a social network.
Horizon has car meets, drop-in/drop-out online play, online road trips. Horizon’s focus is about showing and sharing your car, joining up with others, and having a good time racing in whatever fashion you feel like.