The complete edition includes several DLC missions, including The Consul's Car, A Slip of the Tongue (both now integrated into the main storyline), Nicholson's Electroplating and Reefer Madness.
As well as this there are some control and performance improvements.
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There are 21 missions in LA Noire but you can obtain/buy extra cases. There are also street crimes which are mini-cases, there are 40 street crime cases. The Complete Edition has all the pre-order and DLC cases, there are 26 in total, in full.
You can free roam in LA Noire at any time, but if you do so during cases, it will affect your rating if you cause chaos and damage. If you want to truly free roam, complete one of the ranks and you will unlock a cost free roam. For example, complete Traffic Desk and you will get to free roam in that rank. Complete Homicide Desk and you will unlock free roam in that rank, etc.
Some of the dead bodies are naked.
If I'm correct it takes place during the 1940s or 50s.
Because The Following: Blood and Gore Nudity Sexual Themes Strong Language Use of Drugs Violence