If you save the date in a proper datetime/timestamp field, then you CAN compare them, and mysql (and any other database as well 🙂) should "do the right thing".
i.e.:
create table test (dt date);
insert into test values ('2003-04-03');
insert into test values ('2003-04-06');
insert into test values ('2003-10-08');
insert into test values ('2002-05-23');
insert into test values ('2001-05-23');
select * from test where dt >'2003-04-04');