Bill Clinton claims to be a Christian, so who can say if anything would have been different if he was a Christian?
Any believer in Christ, who turns away or lapses in their faith, also has the ability perform as antichristian behavior and thinking as possible.
Becoming Christian once, doesn't mean that person will no longer be immoral or even miss the mark by God's standards. Instead, the significance of Christianity is that it is what God provides to overcome sin, and still have a righteous, holy relationship with God on His terms.
Judgment and righteousness are preserved in Christianity, without corrupting either. By walking the Christian way of life, past, present, and future sins have already been redeemed by Christ, provided for by God to us, allowing us to face Him through faith in Christ, and then also have forgiveness in our relationship with Him.
Redemption and forgiveness from his wife is another matter. Through faith in Christ, that also may be healed.
I suspect that meaning was reflected in Brit Hume's statement.
Bill Clinton claimed to be a lot of things, and if I claim to be a pirate, that doesn't make it so, either.