Harry's bravery and courage in some cases are exemplary. This extent of that quality is not shown in Edward along with many other aspects of Harry.. If this is so how can we compare Harry with Edward...

Because they have other aspects in common.

-Edward and Harry are both the main male characters.
-They both left their loves to keep them safe.
-Neither of them develop very strong ties outside their small social circle.
-They both are "adopted," after loosing both their parents, by much larger families.
-They both save the girls they end up falling in love with and marrying (Harry saves Ginny in CoS and Edward saves Bella from James in Twilight).
-Both are ostracized at one point throughout the series (Edward was in the beginning of Twilight and Harry was in OotP).

Yes, of course Edward can never be compared to Harry. Harry accepted, even embraced, the possibility of death, whereas Edward escaped death at someone else's cost and became an immortal vampire. Edward is incredibly strong and fast, whereas Harry, being human and ordinary, with no extraordinary talent or intellect like Voldemort, stood up for justice, endured so much of torture, even the Cruciatus, went through so much and still didn't break down, faces numerous challenges, and still survived, and accomplished the big mission for which the entire wizarding world relied on him. Moreover, Edward had eyes only for his girlfriend, whereas Harry broke up with Ginny because he had a more important and more dangerous journey to travel alone, and save the wizarding community. And last but not the least, Harry has a pulse. :)

Other aspects? I will return later and elaborate.

I really love both series. You cannot compare Harry and Edward. Edward is a romantic lead. Harry is a reluctant hero. Two completely different genres.