"Flag" row of duplicate entries with the greatest date

Your dates are entered as text so the formula doesn't work. Here, I converted them to numbers.

Alternately, you can modify the formula to

=IF(COUNTIF($E$10:$E$1509,E10)>1, IF(MAX(($N$10:$N$1509)*($E$10:$E$1509=E10))=N10+0,"X",""),"X")
...confirmed by pressing CTRL+SHIFT+ENTER to activate the array, not just ENTER. You will know the array is active when you see curly braces { } appear around your formula. If you do not CTRL+SHIFT+ENTER you will get an error or a clearly incorrect answer. Press F2 on that cell and try again.