What's the difference from a mere duplicate removal functionality point of view
Apart from the fact that unlike DISTINCT, GROUP BY allows for aggregating data per group (which has been mentioned by many other answers), the most important difference in my opinion is the fact that the two operations "happen" at two very different steps in the logical order of operations that are executed in a SELECT statement. 
Here are the most important operations:
- FROM(including- JOIN,- APPLY, etc.)
- WHERE
- GROUP BY(can remove duplicates)
- Aggregations
- HAVING
- Window functions
- SELECT
- DISTINCT(can remove duplicates)
- UNION,- INTERSECT,- EXCEPT(can remove duplicates)
- ORDER BY
- OFFSET
- LIMIT
As you can see, the logical order of each operation influences what can be done with it and how it influences subsequent operations. In particular, the fact that the GROUP BY operation "happens before" the SELECT operation (the projection) means that:
- It doesn't depend on the projection (which can be an advantage)
- It cannot use any values from the projection (which can be a disadvantage)
1. It doesn't depend on the projection
An example where not depending on the projection is useful is if you want to calculate window functions on distinct values:
SELECT rating, row_number() OVER (ORDER BY rating) AS rn
FROM film
GROUP BY rating
When run against the Sakila database, this yields:
rating   rn
-----------
G        1
NC-17    2
PG       3
PG-13    4
R        5
The same couldn't be achieved with DISTINCT easily:
SELECT DISTINCT rating, row_number() OVER (ORDER BY rating) AS rn
FROM film
That query is "wrong" and yields something like:
rating   rn
------------
G        1
G        2
G        3
...
G        178
NC-17    179
NC-17    180
...
This is not what we wanted. The DISTINCT operation "happens after" the projection, so we can no longer remove DISTINCT ratings because the window function was already calculated and projected. In order to use DISTINCT, we'd have to nest that part of the query:
SELECT rating, row_number() OVER (ORDER BY rating) AS rn
FROM (
  SELECT DISTINCT rating FROM film
) f
Side-note: In this particular case, we could also use DENSE_RANK()
SELECT DISTINCT rating, dense_rank() OVER (ORDER BY rating) AS rn
FROM film
2. It cannot use any values from the projection
One of SQL's drawbacks is its verbosity at times. For the same reason as what we've seen before (namely the logical order of operations), we cannot "easily" group by something we're projecting.
This is invalid SQL:
SELECT first_name || ' ' || last_name AS name
FROM customer
GROUP BY name
This is valid (repeating the expression)
SELECT first_name || ' ' || last_name AS name
FROM customer
GROUP BY first_name || ' ' || last_name
This is valid, too (nesting the expression)
SELECT name
FROM (
  SELECT first_name || ' ' || last_name AS name
  FROM customer
) c
GROUP BY name
I've written about this topic more in depth in a blog post