I think by day, you're referring the seven days in a week: Sunday, Monday, and so on. by date you're talking about what particular date of a month: 23rd of April, 1st of August, etc.
thanks for your answer, LeKha, and I have ever thought about that but now, for me, by day we referring the time unit expressed by one circle the earth turns around its exis. for example, one day, two days...and with "date" we are talking about the particular date as time unit of a month: 31 August,...do you agree with me?
My thoughts of difference between them: A "date" referring to a specific time whereas a "day" is not referred as a specific time unless you are saying,"the day before yesterday I was..,on that day..,that day I was..," (in a conversation).