First of all, both approaches are valid and quite useful. A modern martial artist should be able to do both.
Why the divergent approach? It's hard to say because it's so thoroughly a historical-cultural difference rather than a true philosophical choice, which means it often comes down to luck or personal preferences of unknown people in the unrecorded past.
Nevertheless, one explanation that I find persuasive is that the Japanese/Korean karate philosophy is extremely different from Muay Thai judging and its value system.
The former is strongly influenced by a military fetish, thus highly valuing orderly regimentation (especially for large groups) and a somewhat unrealistic elevation of the "one hit one kill" philosophy. This happened to evolve into extremely low-volume, partial-contact competitions largely dependent on reaction time, featuring single explosive attacks that one can argue would have been disabling had the strike not been pulled before impact. Under these conditions the snap kick is an extremely low-risk, high-reward tool.
In contrast, muay Thai is strongly influenced by a very specific tradition of frequent, public, full-contact, one-on-one competition that involved getting paid either directly or through betting. This happened to lead to one particular (logical) strategy: avoiding high-risk, high-damage affairs, because the upside of a bloody fight today isn't justified by the downside of not being able to fight next week and thus missing out on future income.
This in turn happened to create judging criteria that valued demonstrating composure and thus a desire to disrupt the composure of one's opponent. Forcing your opponent to stumble or limp thus became a high-value tactic. Two logical tools to reach for in this case are a strong round kicks (whether to the thigh muscle, body, or head) and a push kick, both of which we see in abundance in muay Thai. These strikes are quite good at forcing the opponent out of their stance when the whole point is to no-sell every attack.
(An aside: the Okinawan karate tradition actually contains both kicks, though its teep-like push kick has fallen out of popularity in proportion to the degree it borrows from the Japanese-style large-group approach.)