All of the answers above are correct, for the case of DISTINCT on a single column vs GROUP BY on a single column.
Every db engine has its own implementation and optimizations, and if you care about the very little difference (in most cases) then you have to test against specific server AND specific version! As implementations may change...
BUT, if you select more than one column in the query, then the DISTINCT is essentially different! Because in this case it will compare ALL columns of all rows, instead of just one column.
So if you have something like:
// This will NOT return unique by [id], but unique by (id,name)
SELECT DISTINCT id, name FROM some_query_with_joins
// This will select unique by [id].
SELECT id, name FROM some_query_with_joins GROUP BY id
It is a common mistake to think that DISTINCT keyword distinguishes rows by the first column you specified, but the DISTINCT is a general keyword in this manner.
So people you have to be careful not to take the answers above as correct for all cases... You might get confused and get the wrong results while all you wanted was to optimize!