One of the scenarios I found useful to use CTE is when you want to get DISTINCT rows of data based on one or more columns but return all columns in the table. With a standard query you might first have to dump the distinct values into a temp table and then try to join them back to the original table to retrieve the rest of the columns or you might write an extremely complex partition query that can return the results in one run but in most likelihood, it will be unreadable and cause performance issue.
But by using CTE (as answered by Tim Schmelter on Select the first instance of a record)
WITH CTE AS(
SELECT myTable.*
, RN = ROW_NUMBER()OVER(PARTITION BY patientID ORDER BY ID)
FROM myTable
)
SELECT * FROM CTE
WHERE RN = 1
As you can see, this is much easier to read and maintain. And in comparison to other queries, is much better at performance.