UPDATE a column by COUNT of another table
Same query. different join syntax.

The first is old style ANSI, the latter is later ANSI. If you must pick between them, pick the latter.

I wouldn't use either of them due to the limitations of USING (can't be aliased). Instead write as:

UPDATE table1 A 
INNER JOIN (SELECT id,COUNT(*) idcount FROM table2 GROUP BY id) as B
  ON B.id = A.id 
SET A.Freq = B.idcount


See this Oracle answer as to why USING is not to be preferred: https://stackoverflow.com/questions/456684/mixing-using-and-on-in-oracle-ansi-join

And here is a similar discussion for SQL Server: https://stackoverflow.com/questions/1599050/ansi-vs-non-ansi-sql-join-syntax

In general, get used to writing SQL syntax in a way that is portable. While you may never port your code, you will at least be able to port your skills around to multiple database (Which is generally a cool thing to do to avoid becoming a one-trick pony).