Basically, when your FROM clause lists tables like so:
SELECT * FROM
tableA, tableB, tableC
the result is a cross product of all the rows in tables A, B, C. Then you apply the restriction WHERE tableA.id = tableB.a_id
which will throw away a huge number of rows, then further ... AND tableB.id = tableC.b_id
and you should then get only those rows you are really interested in.
DBMSs know how to optimise this SQL so that the performance difference to writing this using JOINs is negligible (if any). Using the JOIN notation makes the SQL statement more readable (IMHO, not using joins turns the statement into a mess). Using the cross product, you need to provide join criteria in the WHERE clause, and that's the problem with the notation. You are crowding your WHERE clause with stuff like
tableA.id = tableB.a_id
AND tableB.id = tableC.b_id
which is only used to restrict the cross product. WHERE clause should only contain RESTRICTIONS to the resultset. If you mix table join criteria with resultset restrictions, you (and others) will find your query harder to read. You should definitely use JOINs and keep the FROM clause a FROM clause, and the WHERE clause a WHERE clause.