If you can't figure out how to do something with tar, you can try pax, which you can think of as a modern version of tar (and cpio). It can do most of what tar can do (at least as long as tape drives aren't involved), and its syntax doesn't suffer from decades of cruft.
In particular, pax offers a
-T [from_date][,to_date] option, so you can do something like
pax <backup.tar -r -pe -T 1006150000 to extract only files dated June 15 or newer. Alternatively,
pax <backup.tar -r -pe -u will refrain from extracting files for which the version in the archive is older than the version in the extraction directory.
Pax is often not installed by default, but all linux distributions should have it. It's specified by Single Unix, so it should be available on all unix systems and unixy environments such as Cygwin.
Finally, if you want really fine control, you can extract the files another location and use
rsync's advanced possibilities, or
find to fine-tune what you want to copy (perhaps with
pax -rw -pe...)