There is a general pattern that writing first appears as pictograms, and steadily transitions into logograms. The early major three were Sumeric cuneiform (no known modern descedants), Egyptian hieroglyphs (ancestor of all Western alphabet systems via the Phoenician and Greek alphabets), and the Chinese characters (supplemented by syllabaries and alphabets in Japan and Korea).
Why did the East Asian cultures, and in particular China, continue using a logographic system into the modern age, while Western Asia relatively very early on switched to alphabets? What is it that happened in Phoenicia and Greece to make it so? Why not in the East?
supplemented by syllabaries and alphabets in Japan and Korea
…and Vietnam.
This is asked somewhat often on /r/linguistics, though I'm having trouble digging up my previous answer. I'll try to summarise, and then if you have more questions about specific points I'll be happy to go into more detail.
Part of the issue here is you're really asking multiple questions which aren't as necessarily connected as you might think.
Why Chinese languages still use characters today? That's the simplest answer. They're still in use because there's no reason or motivation to change. Without a social or politically motivating factor, there's been no reason to change the established and fully functioning system, and the things that would gradually change it haven't had that effect due to factors such as prestige that the characters enjoy.
Why did it change in Japan, Korea, Vietnam? In Korea, the change to the current alphabet took centuries. While it was invented centuries ago, it wasn't until the last 100 or so years that it because truly widespread, and then not until the 1990s that newspapers and academic texts more completely abandoned them, though they do still see some use.
The third question is what caused the change in the West, which is really the same thing that caused Japanese kana to come about. Frequency of use and transmission to other areas, all over time, will naturally result in changes over time. This happened with Chinese a few thousand years ago before forms were standardised and those standards enforced. In the case of the West, it's not that there was some big event or social upheaval that lead to a change in the writing system. It's simply that there wasn't the same sense of orthographic prestige tied to a script, and there was also not the same sort of cultural enforcement of the use of said script over a large area for a period of centuries and centuries.
Unfortunately the answer is a bit more boring than you might have hoped, but it ultimately just boils down to slow moving cultural factors and not major historical events.