When is a good situation to use a full outer join?

It's rare, but I have a few cases where it's used. Typically in exception reports or ETL or other very peculiar situations where both sides have data you are trying to combine.
The alternative is to use an INNER JOIN, a LEFT JOIN (with right side IS NULL) and a RIGHT JOIN (with left side IS NULL) and do a UNION - sometimes this approach is better because you can customize each individual join more obviously (and add a derived column to indicate which side is found or whether it's found in both and which one is going to win).

I noticed that the wikipedia page provides an example.

For example, this allows us to see each employee who is in a department and each department that has an employee, but also see each employee who is not part of a department and each department which doesn't have an employee.

Note that I never encountered the need of a full outer join in practice...

Just today I had to use Full Outer Join. It is handy in situations where you're comparing two tables. For example, the two tables I was comparing were from different systems so I wanted to get following information:

1. Table A has any rows that are not in Table B
2. Table B has any rows that are not in Table A
3. Duplicates in either Table A or Table B
4. For matching rows whether values are different (Example: The table A and Table B both have Acct# 12345, LoanID abc123, but Interest Rate or Loan Amount is different

In addition, I created an additional field in SELECT statement that uses a CASE statement to 'comment' why I am flagging this row. Example: Interest Rate does not match / The Acct doesn't exist in System A, etc.

Then saved it as a view. Now, I can use this view to either create a report and send it to users for data correction/entry or use it to pull specific population by 'comment' field I created using a CASE statement (example: all records with non-matching interest rates) in my stored procedure and automate correction, etc.

If you want to see an example, let me know.

I've used full outer joins when attempting to find mismatched, orphaned data, from both of my tables and wanted all of my result set, not just matches.