The two terms come from the design of reinforced concrete. The old way, Working Stress, (similar approach to ASD with steel). The service load is the best estimate of the actual load that a concrete member may be called on to support.
The current way, Ultimate Strength. The design load is the service load increased by specified load factors in order to provide a factor of safety.
Traditional Examples:
Service Load = Dead Load + Live Load
Design Load = 1.4 x Dead Load + 1.7 x Live Load
Reference: "Simplified Design of Reinforced Concrete - Fourth Edition", Harry Parker