The Biewer, named after the founder,originated in Germany in the mid 80's. Over the years they are selectively bred and a standard was set. The standard is similar to the Yorkie, but calls for specific color placement. White paws, tip of the tail, symetrical coloring on the head and face, etc and no docking of tail. It has been recognized as a separate breed by some organization and can be shown in some venues. Not in the AKC
Partis are tri-colored Yorkies. They can be registered with but not shown in AKC. Can be shown in some other venues. There is no standard set for them that allows for the non-standard coloring, but typically, most Parti breeders follow the YT standard with color as the exception.
Biewer info can be found here...
http://www.biewerbci.org/
You can get more info on the Parti Yorkie here...
http://www.partiyorkshireterrierclub.com/
I know for a fact that there is is a lot of Biewers out there that have been DNA tested that do not belong to the BTRA.