Kick boxing was "i beleive created" in America "i beleive" by combining boxing with full contact Karate. ( dont quote me though)
Mui thai is a thraditional combat MA from Thailand.
Its a stand up MA that uses kicks knees and elbowsand traditionally boxing was not a big thing but the assimulated it into there style after seeing its effectve use by the Western Boxers.
It is a brutally effective form of MA that is proven in the ring and thus the training is orientated around ring fighting.
Kickboxing is very similar to Mui thai however the stances are a bit diffrent. Strictly speaking a Mui Thai practitioner stands in a squared off stance with both hand raised in front with the knuckles facing the fighters face and the forearms facing towards teh opponent. Kick boxes strictly speaking will use a boxers stance.
Also Mui Thai enfassises on leg kicks and the use of elbows and Knees from the clinch for close combat where kick boxers prefer to keep a "medium" distance and finish an opponent with there fists.
Mui Thai enfasises kicking with the shin i know kick boxing has no real rule or enfasis on how it kicks although i think kickboxing strictly speacking likes to use foot and the junction of the foot with the shin to kick with for added reach more in line with TKD or Karate.
*Disclaimer some of my facts might be incorrect feel free to correct me hehe