Explicit joins should always be used; implicit joins have several problems:

• The join condition is somewhere in the WHERE clause, mixed up with any other filter conditions. This makes it harder to see which tables are joined, and how.
• Due to the above, there is a higher risk of mistakes, and it is more likely that they are found later.
• In standard SQL, explicit joins are the only way to use outer joins:

SELECT d.Name,
       e.Fname || e.LName AS EmpName
FROM      Departments AS d
LEFT JOIN Employees   AS e ON d.ID = e.DepartmentID;

• Explicit joins allow using the USING clause:

SELECT RecipeID,
       Recipes.Name,
       COUNT(*) AS NumberOfIngredients
FROM      Recipes
LEFT JOIN Ingredients USING (RecipeID);

(This requires that both tables use the same column name.  
USING automatically removes the duplicate column from the result, e.g., the join in this query returns a single `RecipeID` column.)