You will earn an extra month of interest! Both I and EE saving bonds are issued the first of the month in which they are purchased. In you purchase them on the last business day of the month, you will receive interest as if you bought them the first day of the month.
However, interest is always paid the first day of the month. If you cash-in your bonds two weeks into the month, you will not receive any further interest payments and will miss re-investing the money for those two weeks. Bottom line, if you buy a savings bond the last business day of the month and cash it in the first business day of the month, you will be getting an addition month of interest for free.
Interest rates for the EE savings bond and the fixed rate component of the I bond are set on May 1st and November 1st. For the EE bond, the rate is based upon (but somewhat lower than) the 10 Year Treasury note. Because you can compare the rate on 10 Year US Treasuries today to the rate six months ago , you can effectively choose the higher rate by either purchasing the savings bond in April or October, or waiting a couple weeks for the rate to change.