Both names refer to the same mixed breed, a cross between a Shi-tzu and a Bichon Feese. The question is, which one is becoming more accepted in the main stream. As far as I can tell the names are interchangable, wiki defines both shichon and zuchon as:
a breed of dog, a cross between a Bichon and a Shih Tzu
American Kennel Club does not recognize this hybrid mix, however the American Canine Hybrid Club does, and calls it a Zuchon.
We have written about the popularity of the terms Zuchon and Shichon this January based on tecnocrati search and at the time the term Zuchon was winning. Looking at this again it seems to be tied.
Finally, checking out google trends the evidence stacks up agains Zuchon in favor of Shichon.
So, next time you are walking your adorable little teddy bear puppy and someon asks what is it... what's it gonna be - Zuchon os Shichon?