Specifically, why was Odysseus's bow hard to string, and why would this bow type not have been used by the English/Welsh army at Agincourt?

Odysseus' bow is thought to have been a composite bow, requiring great skill and leverage to string. Or maybe it is a backsprung bow. Or a composite recurve bow. Homer wasn't all that clear!

https://en.wikipedia.org/wiki/Odysseus
http://education.yahoo.com/homework_help/cliffsnotes/the_odyssey/58.html

The English and Welsh archers at Agincourt used longbows, which are easy to reload and shoot arrows more quickly than other types of bow. maiking possible the release of a 'hailstorm of arrows'.

https://en.wikipedia.org/wiki/English_longbow