When should I make a variable equal to a previous variable?

In this limited example, no there isn't a need to set full_name first, aside from being a teaching tool on how to pass one variable to another.

In a real world example say we have a registration form with a field for first name and another for last name. Once the user registers that data is submitted to a database of users. The user later logs in and gets a welcome message.

In the database, the users table has a column for each of first_name and last_name. To create the welcome message after a successful login the first_name and last_name is pulled from the database and then they are concatenated to form a full name.

Some example super simple code could look like:

// Inside a function somewhere:
// Previously established database connection
// Data is selected based upon username id
$first_name = $Database->Users['username']->FName;
$last_name = $Database->Users['username']->LName;
$full_name = $first_name . ' ' . $last_name;
$welcome_message = 'Welcome ' . $full_name;
// Send $welcome_message to the client

In this example it is recommended to use one variable for each record pulled from the database as it helps with organization as well as making changes easier later on if something changes with the application.

I hope that helps to clarify a little about the benefits of separating data between variables, as well as the benefits of combining variables when needed.