Because dates in access stores info about hours, minutes and seconds (and maybe even milliseconds), if you want to delete records with only same date (year, month and day) try to restrict your query to this:
DELETE FROM Table1 WHERE
YEAR(Dates_Worksheet.Date_Payment) = YEAR(Table1.PaymentDate) AND
MONTH(Dates_Worksheet.Date_Payment) = MONTH(Table1.PaymentDate) AND
DAY(Dates_Worksheet.Date_Payment) = DAY(Table1.PaymentDate)
Hope that helps.