I can provide a little insight about how and why Vietnam adopt a Latin alphabet. As for why its South East Asia neighbor didn't, you may have to wait for someone with more expertise on those nations.

For some context, a latin alphabet has been around since 17th century, used by missionaries to record the local language, but it was circulated mostly among the Christian community - a very small community by far. This latin based script latter developed into the present day "Quốc ngữ" (National Language).
Anyway, prior to 19th century, Vietnamese writing consisted of 2 logographic system.

• "chu Han" which is the standard set of classical Chinese character, it is the formal writing language, used in formal document. Chu Han was also the compulsory language used in the yearly officer exam.
• "chu Nom" is the system that used abbreviated Chinese character to write Vietnamese spoken language. You may check out an explanation of it here
  So if we talking about a 18th century highly educated Vietnamese person, he/she would likely be able to write in both scripts. He/she would be taught a curriculum consist of literature, poetry, philosophy, history under Confucian value; either at his/her home with a private tutor or at the house of a teacher with some other kids in the village. Once grown up, he might decide to take the officer exam (men only) and become a government bureaucrat. He will proceed his work with documents written in Han. And in his free time, he might write a poem or novel either Han or Nom, whichever he prefered. If he failed the exam, a secondary choice of career would be opening his own classroom to teach the same curriculum to the next generation.

Everything changed in the 19th century. From 1858 to 1886 part after part of Vietnam became French colony. Right after establishing full control over South Vietnam in 1878, the French administraion made Quốc ngữ the official writing in all formal documents, complete transition must be made after four years, during which the general use Quốc ngữ was promoted through a more standardized education system aligned with Western value in stead of Confucianism. Same policy was annouced in the North in 1910; and by 1918, the officer exam was canceled, cut off the career prospect of all Confucian-educated scholars. Needless to say, Confucian intellectual opposed the teaching of new scripts but there's not much they can do. By 1930s, a new Western-educated intellectual class had arised. Thanks to the cultural exchange, Vietnamese modern literature was born in this period, booming, written with the Latin scripts while conventional forms of Han-Nom literature faded away. So, Han-Nom has largely been abandoned in favor of Latin based writing for both formal and general use. The final push was made after the communist revolution in 1945, new government headed by Ho Chi Ming immediately announced the education policy "Bình dân học vụ" teaching the Latin scripts in a campaign to fight illiteracy (which was like 95%).

That's the how, now to the why. There were couple of reasons why French administration promoted the Latin scripts. One, the former dual system was just painfully complicated. Two, to train the necessary bureaucrats, Vietnamese bureaucrats were valued highly and served not only locally but was also in Laos and Cambodia. And three, to increase the Western cultural influence and limit the influence from China - which was burning with various revolutionary ideas at the time. And not all Confucian scholars were against the Latin script, even if they are not entirely approval of French influence. Some sent their children to be educated in Western curriculum for better career prospect; the most notable example being Ho Chi Minh who studied Han-Nom in his childhood but latter joined a Westernized high school (which he soon got expelled for protesting against high taxation). The Latin script was also seen as a tool to reform the weak colonized country; free education organization like Đông Kinh Nghĩa Thục was founded by Confucian scholars but taught the new script (until it was shutdown for teaching the populace too many ideas that didn't sit well with colonialism).

Source: Lich su chu quoc ngu, 2007, Nha Xuat Ban Ton Giao