Thought I'd add my own little answer to what the Complete Edition contains. It pretty much adds all of the DLC that has been released for the game and crams it onto the game disk. Here's what it contains:
SOUTH CENTRAL MAP PACK:
A map expansion giving you a new district to play in: South Los Angeles, still sometimes known as South Central.
SOUTH CENTRAL EXPANSION PACK:
An entire expansion adding a lot more content to the game:
New races
New missions
New battle maps
New music tracks
New red light races
New delivery missions
New aftermarket parts
New vehicles, including two all new classes, Lowrider and SUV:
1955 Bel Air (Group 1, Lowrider)
1963 Riviera (Group 2, Lowrider)
1964 Impala (Group 2, Lowrider)
1965 Malibu (Group 2, Lowrider)
Range Rover Supercharged (Group 3, SUV)
Range Rover Sport (Group 3, SUV)
1966 GTO (Group 2, Muscle)
1970 Chevelle SS (Group 2, Muscle)
CLK63 AMG Black Series (Group 4, Luxury)
SOUTH CENTRAL VEHICLE PACK 1:
Contains three new cars:
Audi S5 (Group 3, Luxury)
CL65 AMG (Group 4, Luxury)
Eclipse GT Spyder (Group 3, Tuner)
SOUTH CENTRAL VEHICLE PACK 2:
Contains four new vehicles:
Aston DB9 (Group 4, Exotic)
Cadillac XLR V (Group 4, Luxury)
Ducati 1098R (Group 4, Bike)
Lancer Evolution X (Group 3, Tuner)
In addition to the South Central content, you also get a nice little bonus:
COP CAR PACK:
Enforce the law with police cruiser versions of high performance sedans, with fully functioning lights/sirens and performance upgrades:
1996 Impala SS Police (Group 2, Police)
1996 Impala SS CHP (Group 4, Police)
Charger SRT-8 Police (Group 3, Police)
Charger SRT-8 BP (Group 3, Police)
Charger SRT-8 CHP (Group 4, Police)
So pretty much, the Complete Edition is the standard game with a whole lot more injected into it.