Though both companies had early common origins, they have taken completely paths at least since the end of WW II.
Lange effectively went out of existence, while what is now the GO factory was turned into a State combine called GUB (Glashutter Uhrenbetrieb) under the socialist East German government.
Lange was revived in the 1980's by a group including the recently deceased and much revered Mr. Gunther Blumlein and Mr. Walter Lange, a direct blood descendant of the original Lange founder, with much technical assistance from IWC and Renaud & Papi. A. Lange Und Sohne was later merged into LMH, along with JLC and IWC, and recently was sold as a group to Richmont/Vendome.
GO was privatised by Mr. Heinz Pfeiffer, a West German businessman, when East and West Germany were reunited. GO is now part of the Swatch Group.
GO and Lange have taken rather different approaches to the design and production of their model lines, Lange going the ultra high end route, with perhaps finer cosmetic finishing, and positioning themselves against Patek, AP, VC, the traditional big three Swiss brands.
GO watches have a different feel - a bit less dressy, more functionally utilitarian, though also of high quality. By their pricing and design, it would seem they are going more after the technical market, with interesting complications. Their final finishing is perhaps not quite to the level of Lange, but at their price point, offer surprisingly good value.
Many people confuse Lange and GO, or consider them direct competitors, since they are both generally in the high grade category, and both are located in Glashutte, just down the street from each other. Personally, I don't consider them competitors at all, other than that they both make very nice watches. They are quite different, in feel and philosophy.