Nexuiz was originally developed by Lee Vermeulen and Ashley Hale (Ladyhavoc), who started Alientrap in the summer of 2002. The goal of the project was to create a high quality first person shooter that could be played freely across all platforms in one package: PC, Mac, and Linux. The engine that powered Nexuiz was Forest Hale’s Darkplaces engine.
The first version of the game was released in 2005. Development continued with many online contributors over the years, with version 2.5 released in October of 2009. Since its release it has been downloaded over 6 million times, and is included with many Linux distributions.
In mid-2010 it was announced that a new XBLA, PSN, and Steam downloadable remake of Nexuiz would be done from the ground up by IllFonic using Crytek’s CryENGINE3 game engine. Original Nexuiz designer Forest ‘LordHavoc’ Hale worked with Illfonic on the new version, many of the GPL contributors forked the project into a new game Xonotic, while Lee Vermeulen moved towards Capsized development.
Illfonic’s Nexuiz was released for XBLA in February 2012, and Steam May 3rd 2012.
I believe classic refers to the FOSS version before the fork, while "Nexuiz" refers to the commercial version.