Some people mistaken the Muay Thai Cut Kick with the Sweep. They are two separate kicks, basically the Sweep Kick is kind of tripping the opponent when engaging in a close distance fight. The Sweep would require pushing the opponent, making them off-balance as you are catching their leg or just straight pushing them off balance.
The fine line between both kicks is that they similarly lead the opponent to fall. The difference is that after launching the attack of the Cut Kick, the opponent wont be able to stand.
The Cut kick is more of an intercepting kick that counters an attack and delivers brutal force to the standing leg. It would be similar to cutting down a tree from the trunk, with an axe. It’s delivered with the shin bone not any other part of the leg.
Be careful when using it in a fight, although the Cut Kick is very powerful, it does take an advanced level fighter to perform it. When done by a professional it can demoralize the opponent, as it is done with accuracy, power and speed.
An amateur that is just starting should not focus on the Cut Kick. The dangers that may arise, if they were to lay down their defenses and open their face to attacks, can vary from getting caught with a head kick or even straight up punches.
The video below shows how to perform the Cut Kick. The main purpose is to break down the leg.
I have dissected the varying attacks that you could be exposed to, the punches or the Round Kick like the Head Kick and Middle Kick. Also this was done gently when I performed it as I was trying to avoid any damage to the student. But if you decide to do it in a fight make sure you get your full swing from the leg.