Wikipedia gives a brief overview of how the commandment has generally been seen as "new." It accords with what I've heard through the years:
The "New Commandment", the Wycliffe Bible Commentary states, "was new in that the love was to be exercised toward others not because they belonged to the same nation, but because they belonged to Christ...and the love of Christ which the disciples had seen...would be a testimony to the world".
One of the novelties introduced by this commandment – perhaps justifying its designation as New – is that Jesus "introduces himself as a standard for love". The usual criterion had been "as you love yourself". However, the New Commandment goes beyond "as you love yourself" as found in the ethic of reciprocity and states "as I have loved you", using the Love of Christ for his disciples as the new model.
The idea that Jews were only commanded to love one another, and not to love outsiders, doesn't seem to hold water. Sixteen verses after "love your neighbor," we read:
The foreigner residing among you must be treated as your native-born. Love them as yourself, for you were foreigners in Egypt. I am the LORD your God. (Leviticus 19:34 NIV)
Not to mention the fact that according to Luke's gospel, Jesus had already claimed that "my neighbor" in the context of the commandment is whomever you can show mercy to. So if you believe that John and Luke report the same Jesus, as Christianity has traditionally taught, that can't be what's "new" about Jesus' New Commandment.
John Piper helpfully expounds on what makes the New Commandment new in a sermon on John 13:
First, the command is new because it is a command to live out the love of Jesus. Second, the command is new because it is a command to live on the love of Jesus. The words “as I have loved you” contain a pattern for our love for each other, and they contain a power for our love for each other.
Loving each other is not a new command per se. It was already there in the Old Testament (Leviticus 19:18, “You shall love your neighbor as yourself”). What’s new is that Jesus is now the pattern we live by and the power we live on. Let’s look at these two kinds of newness.
Piper quotes Jesus' words from earlier in John 13:
Do you understand what I have done to you? You call me Teacher and Lord, and you are right, for so I am. If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another’s feet. For I have given you an example, that you also should do just as I have done to you.
What this seems to mean is that Jesus' disciples are to serve one another, regardless of their social status. Piper thus says that the "pattern we live by" according to the New Commandment is to "lay aside status and rank and prestige and privilege and take the form of a servant" and engage in "practical deeds of helpfulness."
So, "Love your neighbor as yourself" seems to involve loving everyone. "Love one another as I have loved you" seems to involve loving fellow believers. Obviously this is a subset of the old command, but it raises the standard, from "as you love yourself" to "as I (Jesus) have loved you," and it gives a new purpose, that "by this everyone will know that you are my disciples." This purpose is reminiscent of Jesus' words in the Sermon on the Mount: "Let your light shine before others, that they may see your good deeds and glorify your Father in heaven."
It doesn't appear that the old command was done away with either, since Paul says in a discourse on love:
Do not repay anyone evil for evil. Be careful to do what is right in the eyes of everyone. If it is possible, as far as it depends on you, live at peace with everyone. Do not take revenge, my dear friends, but leave room for God’s wrath, for it is written: “It is mine to avenge; I will repay,” says the Lord. On the contrary: “If your enemy is hungry, feed him; if he is thirsty, give him something to drink. In doing this, you will heap burning coals on his head.” Do not be overcome by evil, but overcome evil with good.
So both commands are in effect, but they have different scopes and different purposes.
Love your neighbor as yourself
a. Command: love all people
b. Standard: love as you love yourself
c. Purpose: to pursue peace on earth
d. In effect: yes
Love one another as I have loved you
a. Command: love and serve fellow believers
b. Standard: love in the ways that Jesus demonstrated
c. Purpose: that non-believers will take notice
d. In effect: yes