1. You often hear that Constantine made Christianity the only allowable religion in the Roman Empire, that he squashed Gnosticism as a Christian possibility. This is false. Constantine made Christianity legal (it had been the target of persecution previously) in AD313. He did not make it the official religion of the Roman Empire. He even patronized some other gods.
Another thing I hear often is that the New Testament canon--the list of authoritative books in the Bible--was determined at the Council of Nicaea in 325, called by Constantine. This is also false. The contents of the NT had mostly been agreed on for years by then, but the final agreement on the edges (e.g., 2 Peter, 3 John) would not really come into play until decades after Constantine's death. There was no universal council. It just more or less happened. In 398, a Western council did affirm these books.
It was a later emperor, Theodosius who in 380, declared Nicene Christianity the only legitimate religion of the empire. A "heterodox" bishop in Constantinople was removed. Then in 381, the Council of Constantinople finalized the Nicene Creed. Only Nicene Christianity, at that point, could be considered catholic Christianity.
2. So the main thing that changed was the end of support for polytheism and the worship of other gods. I believe my colleague David Riggs would tell you that much of polytheistic worship continued nonetheless and went underground. I remember dabbling in an anti-catholic book as a teen that suggested that the cult of the saints was largely the polytheistic religion of the majority gone underground. (no idea whether there is any truth there at all)
3. Did the moral climate of Rome change? I'm not sure how much evidence there is for this. Anything associated with the Roman gods or other gods was outlawed. Magic was outlawed. The Olympic games were outlawed. In 529 Plato's Academy was closed.
In keeping with the Mediterranean worldview, the main thing that changed was religious practice. Rituals and ceremonies associated with Jupiter or Mars were outlawed. The Vestal Virgins were disbanded. The temples associated with these gods would now either fall into disrepair or be re-purposed. Instead of Saturnalia, we would now have Christmas.
Now basilicas of worship would be built. The biblical text would become more or less standardized (thus the Greek behind the King James Version). Doctrine was standardized (Nicene)--Manichaeans for example would be persecuted.
4. So there were changes in religious worship and changes in religious belief. But the Roman empire was not Pietist. The empire did not necessarily become more loving or more biblical in the most meaningful sense of that word. I believe Constantine did make some changes that curbed the practice of infanticide. I would be interested in knowing other changes that took place.
I would argue that the number of "real" Christians probably stayed about the same, just like today, a minority. See Ed Stetzer's claims in more than one place that Christianity today is not in decline in America. It's only that nominal Christians, who arguably weren't really Christians anyway, are no longer identifying themselves as Christian.
5. So when did Europe become "Christian" in anything other than matters of external form and cognitive assent? The next candidate in Western storytelling, I suppose, is the Protestant Reformation.