Checking the source code of MySQL 5.6 CAST() and CONVERT() calls to the same internal function Item_date_typecast, DATE() calls to Item_date_typecast too.
In conclusion there is no difference between DATE(expr) function and the CAST(expr AS DATE).
Refs:
https://github.com/mysql/mysql-server/blob/5.6/sql/sql_yacc.yy
https://github.com/mysql/mysql-server/blob/5.6/sql/item_create.cc