Many reasons are given for the Chinese aversion ot using voice mail in an article in the Wall Street Journal in Jan 2006:
Most people use mobile phones and never turn them off even in meetings, movies and funerals.
Text messaging is the preferred substitute for voicemail messages.
Most are away from their desks in traditional face to face meetings and don't expect anyone to be around to answer their office phone or check messages.
Don't expect to leave a message or consider it a loss of face to do so.
Anyone of importance will have a secretary to answer the phone and take messages.
Less attention is paid to why Americans like it so much. The article seems to imply that Europeans like it equally, which I would dispute. The only reason given is that Americans travel a lot.