Instead of missions, you'll have to solve cases. The "Complete Edition" adds all of the previously released downloadable content cases, including: the "Reefer Madness" and "Then Naked City" Vice cases, the "A Slip of the Tongue" and "The Consul's Car" Traffic cases, and the "Nicholson Electroplating" Arson case.
The "L.A. Noire -- Complete Edition" would be the best fit for gamers seeking an immersive crime-solving experience set in post-WWII Los Angeles.