Why do European languages use a similar alphabet, but South East Asian languages do not?

The answer to this question is fairly straightforward along the lines of historical developments and cultural influences and goes pretty much along the lines offered by jamesqf's answer.

However, a much more illuminating answer would focus on the problematic nature of the assumptions behind the question.

First, Europe and South East Asia are not necessarily the best categories for this sort of comparison.

When it comes to writing systems, you may be better off talking about Mediterranean-Atlantic cultural sphere. Here, you will find at least five different (albeit related) writing systems: Latin, Greek, Cyrillic, Arabic, Hebrew currently in wide usage (millions of users across multiple languages). They can all trace their origins to similar roots but ultimately they underwent independent processes of development and cultural spread.

On the other side you have at least two historical spheres of cultural influence: Brahmic (Indian languages, Tibetan, Thai, Lao) and Sinic (Chinese, Japanese, Korean, Vietnamese). So, talking just about South East Asia as one area is misleading.

If you only focus on the artificial concept of 'modern Europe', you will find that the prevalence of the Latin script is due mostly to its mono-cultural origins of learning in Western Christendom. The surface dominance of the Latin alphabet is only about 200 years old coming on the heels of mass literacy in the West spurred by industrial revolution (to simplify things greatly).