An assignment is a task given to a specific person or group to complete. It can also mean the act of assigning. In some legal fields it can refer to the transferring of ownership of property.
An assignation is the act of assigning or the actual assignment. But it also means a secret rendezvous for lovers, most especially for affairs or illicit relationships. Sometimes this is used to mean a secret meeting or a regularly set meeting, but it would be confusing in this sense to those who are familiar with the standard meaning (see the example below). As a mass noun it can refer to something as being owned or belonging to something else.
Examples
The choice has not been made, but it appears the team is more likely to option Orlando to Omaha, rather than designate Dyson for assignment. [The Kansas City Star]
According to a criminal complaint, Ms Dale, 21, told a detective that Ethan refused to complete the assignment for the first time on 24 February. [International Business Times]
I vaguely remember seeing images of Delhiites being water-hosed by police while rioting for “Nirbhaya,” an assignation that means “Fearless One,” given to Pandey by the media as authorities would not release her name. [Huffington Post]
The rare exceptions include a scene in which Shulem follows his wife to an assignation with Félix, slapping his romantic rival to the sidewalk like a petulant schoolboy. [The Washington Post]
Over the past few months, RPG Enterprises chairman Harsh Goenka, has been having an assignation every other Friday for a few hours. [India Times]