Muay Thai teaches simple and straightforward fighting techniques which are effective in a real fight. Thai kickboxing doesn't focus on some fancy moves that are not very useful in combat.
Muay Thai puts a great pressure on conditioning: stamina and strength training, shin conditioning and core conditioning. That gives you a physical edge over the potential aggressor.
In Muay Thai the main method of learning is by sparring. That means you will be encouraged to spar almost right from the beginning of your training.
You will have to face a real opponent, and learn from him, instead of punching the air. That gives you a great mental advantage over the potential attacker.
I have created a post: Muay Thai for self-defense to fully cover this subject.
In street fight, you have to be able to move around freely and be able to defend against more than just one opponent. You need to be able to look around, evade, and react quickly.
You don't want to get stuck in a situation in which you are sitting on a guy and holding him down on the ground. In that situation some other guy may get involved and soccer-kick you in the face.
What works in the octagon doesn't necessarily work on the street. That is why a stand-up fighting style like Muay Thai is much more effective for the street than a grappling style like BJJ [Brazilian Jiu-Jitsu].
Muay Thai utilizes 5 main sets of techniques:
There are quite a few of them and most of them are very spectacular:
Also: Who was the founder of Muay Thai?
The predecessor of Muay Thai style called Muay Boran can be traced to XVI century Siam. It was developed as a deadly martial art which could be used on a battlefield.
Muay Boran wasn't limited by any rules. The sole purpose of that fighting style was to kill or maim the enemy soldiers using any technique available.
It was an aggressive and very effective fighting style. A lot of Muay Boran fighting techniques can be still observed in modern Muay Thai.
I wrote more about this subject in History of Muay Thai