What is the primary difference between ArcGIS Desktop and ArcGIS Server?
ArcGIS Desktop can work standalone and can also consume GIS services.
ArcGIS Server provides GIS services.
I found the following at a no longer reachable resources.arcgis.com page:
ArcGIS for Desktop — Comprehensive software used by GIS professionals on Windows PCs for the full range of GIS activities including data compilation, mapping, modeling, spatial analysis, and geoprocessing. ArcGIS for Desktop extensions provide additional functionality such as amazing 3D visualization, network analysis, schematics, and geostatistics. Desktop is the starting point and foundation for deploying GIS in organizations. It is used by professional GIS staff for desktop mapping, reporting, and analysis. It is also used by data compilation staff to create and maintain critical foundational data layers that fuel other GIS applications. This role for data stewardship is quite significant.
ArcGIS for Server — Powerful GIS back-office software that enables centralized, enterprise-level geodatabase management and server-based publication of maps and geographic information services throughout the enterprise and on the Internet as web services. ArcGIS Server supports the leading enterprise database management systems (DBMS): Oracle, SQL Server, DB2, Informix, and PostgreSQL. It is available on Windows or Linux servers on-site or in cloud configurations. ArcGIS Server provides the core technology for implementing large-scale GIS in organizations and businesses worldwide.
While PolyGeo's answer is completely correct,I'll try to explain it in a different way.
ArcGIS Server is a Server software, which responds and fulfills requests from clients via several different kinds of services.
ArcGIS Desktop, is more of a client(i.e. desktop) software, which is used in the creation, editing, analysis etc of GeoSpatial Data. It can be used in conjunction with ArcGIS Server. It can consume the services from an ArcGIS Server, as well as used to manage and publish new services.