I'm unable to tell you exactly why neon signs became so popular in East Asian city-scapes, but I can briefly discuss when neon signs became popular in Japan and China (and a few other places).

First of all, the use of neon signs (aka sculpted cathode discharge tubes) commercially for adversising and signage date back to the early 20th century, but speaking in very broad generalities, peaked in popularity in the United States in the period 1920-1960.

I don't a ton of great source to support this, but the wide-spread use of vertically-oriented neon signs seems to have started mainly in Tokyo and Osaka sometimes in the 1960s, and spread to Hong Kong by the end of that decade. By the 1980s, lighted signage had become an essential part of what people considered a "modern" and "developed" urban area. As other parts of Asia (China, S.K., Vietnam) became more developed in the 1980s and 1990s, they copied these regional trends in urban architecture and visual design, since a well-lit cityscape signified modernity, progress, and prosperity.

Sources, or the closest thing I can find to them:

http://www.asianhistoryblog.info/2012/11/a-history-of-asias-neon-signs.html

http://blogs.wsj.com/scene/2014/03/20/hong-kongs-neon-signs-get-art-show/

Also, I haven't read it, but apparently someone did write a history of neon lights and their cultural significance: Christoph Ribbat's Flickering Light: A History of Neon publised in English in 2013 by UCP. link: http://press.uchicago.edu/ucp/books/book/distributed/F/bo15582726.html