Getting Biblical in Daily Life
Jewish Perspective
Considering that Ezra and Nehemiah are both part of the Hebrew Bible, and that they deal with the reconstruction of the central location of ancient Israelite religion—the temple—both books and characters have a central place in Judaism. These books are proof-texts in that they demonstrate the fulfillment of all the earlier prophecies that Judah will eventually be brought back from exile and resettled in the land promised to them.
The ancient rabbis believed that if Moses hadn't existed, Ezra could've fulfilled the same role as receiver of the tablets of the law. He was that devout. The Jewish sages of the first few centuries CE expanded Ezra's importance to include the founding of the Great Assembly, the Sanhedrin, the editing of most of the books of the Torah and prophets, and a host of other practices. Check these out:
The [following] ten enactments were ordained by Ezra: That the law be read [publicly] in the Minha service on Sabbath; that the law be read [publicly] on Mondays and Thursdays; that Courts be held on Mondays and Thursdays; that clothes be washed on Thursdays; that garlic be eaten on Fridays; that the housewife rise early to bake bread; that a woman must wear a sinnar; that a woman must comb her hair before performing immersion; that pedlars [selling spicery] be allowed to travel about in the towns, He also decreed immersion to be required by those to whom pollution has happened. (Source)
Shmoop editors, after exhaustive research, were still unable to find any mention of garlic in the book of Ezra. Grapes, yes; garlic, nada. Regardless, the guy got credit for everything.
Most contemporary Jews would probably acknowledge Ezra as the reviver of Israel's religion who used the Law to re-establish a sense of distinctive identity among the Jews during a very unstable and critical time in Jewish history. He taught the community, not just the priests, and provided a foundation for a Jewish future based on a common set of laws and practices.
As for Nehemiah, the rabbinical tradition also views him as being a good governor and an important part of the revival of Judaism after the exile. However, they also censured him for boasting about his good deeds and for disparaging leaders who came before him in the exile, one of whom was Daniel of Lion's Den fame. This, according to these rabbis, anyway, is part of the reason why Nehemiah isn't really a book of its own, but was originally part of Ezra's book (source). Take that, Nehemiah.
Christian Perspective
The books of Ezra and Nehemiah are of less interest to Christianity, not that Christians don't accept those books as being an important part of the Biblical Canon. But for Christians, the rebuilding of the Second Temple isn't really the beginning of the new covenant, as it is for Jews. While it was assuredly a good thing, it's kind of a false start, in a Christian view. The Second Temple isn't the culmination of anything, and in the Gospels, Jesus predicts its destruction (Mark 13). The death and resurrection of Jesus create what for Christians is the truly new covenant. Details of the temple sacrificial cult were not of much concern to Christian theologians except as it indicated faithfulness to God.
Most Christian commentary on the books addresses the themes of persistence in the face of challenges to the faith, God's meeting our needs, God's protection and encouragement of his faithful, etc. Some see the return of the exiles as representing Christ's redemption of the spiritually exiled. There's the implication that since one of the nation's major goals was fulfilled—the restoration of the Temple—then people can be confident that the ultimate fulfillment will also come with the arrival of Christ.
Islamic Perspective
Muslim authorities don't really have much to say about the details of the actual books of Ezra and Nehemiah, since they deal with so many features that are more uniquely important to Jewish history. However, Ezra is considered a prophet by some Muslim scholars, although others challenge this view.
The Qur'an claims that Jews during the prophet Muhammad's time said that "Uzair" was the "Son of God," whereas Christians claimed that Jesus was. The Qur'an argues that there are no "Sons of God," in the commonly accepted sense of the term, but only prophets. Some Muslim authorities claim that "Uzair" was really the same as Ezra, and was in fact a prophet (source). (For the record, the overwhelming majority of Jews would likely object to the claim that they believed Ezra was the "Son of God.")
According to Islamic tradition, the tomb of Ezra is located in Iraq near the Tigris River. He supposedly died there at the ripe old age of 120 while riding through on his donkey (source).