In Kickboxing front kick, the same side hand does Not go down. In Muay Thai front kick the hand does go down. What is the reason for this difference?
I think this comes down to different cultures due to scoring systems and should not be generalised.
In kickboxing, it is actively encouraged to keep your guard up at all times, including when throwing a kick. That is because you want to minimise risk as hits taken adding up to dizziness is what is to be avoided at all costs and depending on the rules, even light hits may count as a score against you.
In Muay Thai, on the other hand, a display of dominance is your main goal. Accordingly, fighters more often consciously let their guard down in order to show 'look, I don't even need to protect myself against this guy'. Also, if you are not actively trained to keep your hands in front and up, this is the more natural thing to do to keep your balance.
Thus, it comes down to slightly more power with the help of this counter-movement of the arm vs. slightly more defensive capabilities with a hand out ready for block/parry. I wouldn't say that these differences generally hold between these two sports though, as the emphasis on power or safety respectively isn't tied to the sport as such, beyond the mentioned cultural difference.
Accordingly, you will find both in pictures and technical descriptions of both sports, although Muay Thai that is explicitly trained in the context of MMA tends to have their guard/hand up as well, as goals and demands are similar to full contact kickboxing.
I think you will find that it is simply a stylistic difference. Muay Thai tends to drop the arm and use it to pump in the opposite direction for a few of their kicks, most notably the front kick and the roundhouse.
When I did some (limited) training in muay thai I was told that this pump action in the opposite direction to the kick is to increase power, however based on experience I don't believe the movement itself does add power - it tends to help balance more than anything. As you'll see from kickboxing and karate, the pump motion is unnecessary and you can generate significant power without it, although it does take considerably longer to master the technique. So it could be argued that using the pump motion assists with balance enough that beginners or junior students can increase the power of the kick at a faster rate with less training.