I am not entirely sure why you are having that problem but here is an idea.
The default date system in Excel for calculating serial numbers for dates
(which is how Excel operates) starts on Jan 1st 1900. However, you can also
opt to use the 1904 date system. Given that the discrepancy in your dates is
4 years i.e. 2004 becomes 2008 it could be that the worksheet you are pasting
to is set to the 1904 date system and so the serial number is being
interpreted incorrectly.
To check, in Excel...
<Tools><Options...> and under the <Calculation> tab make sure that the <1904
Date System> is unchecked.
The 2nd workbook is set to use the 1904 date system. Change that at
Tools/Options/Calculation. Be aware that if you do that, any dates that exist
on the worksheet will be low by 1462 days. To fix that, you can type 1462 in
an empty cell, edit/copy that cell, then select the cells with the bad dates
and edit/paste special and select the add and values options.