The word "new" is translated from the Greek word kaine. This is interesting because, while it does mean "new" in terms of time, the emphasis in the use of the word, when compared to something of the same kind, in this case, covenants, is on quality - not time. Hence, the emphasis in the use of kaine is on this covenant being better rather than being newer.
In Jesus' Parable of the Old and New Wineskins, kaine also appears. Using this understanding of kaine, the difference between the wineskins is not necessarily age (though that is implied) but quality. One wineskin is dried and cracked, while the other is supple and resilient. Though it may also be newer, it is decidedly better.
Putting this into a modern context, we can make a comparison between a 1910 automobile and a 1995 automobile. The 1995 automobile is a continuation of the same general kind as the 1910 automobile. Both have the same necessary parts: engine, wheels, steering wheel, seats, transmission, brakes, lights, and a nut behind the wheel. But the 1995 model has made the 1910 model obsolete as a viable mode of transportation.
So it is in the comparison between the Old Covenant and the New. Both have the same necessary parts, so that they may be considered of the same "kind," but the New Covenant is so much better and has so much more going for it that it has made the Old one obsolete.
Is there a difference between a testament and a covenant? The word "testament" does not even appear in English translations of the Old Testament, but it appears thirteen times in the New Testament. The Greek word is quite interesting because it does not even mean "covenant" as we think of it. In fact, researchers have been able to find only one usage outside of the Bible—in classical Greek—in which this word is used in the same way that the English and the Hebrew words are. The Greek word is diatheke, and it is the equivalent of our English word "testament" or "will"—not "covenant."
A covenant is an agreement between two parties. The emphasis in on the words "agreement" and "parties." However, a diatheke is a testament or will. As in English, it is a unilateral—a one-sided—declaration of the disposition of property that a person makes in anticipation of his death. Before we die, we usually draw up a declaration of what we want done with our property, and most people do not consult with the people they want to leave their possessions to. It is usually a private matter.
Paul used this singular word—diatheke—where two different words normally would have been used. The interesting thing is that the Greeks have a word for a covenant, suntheke, "a bilateral agreement," but the apostle did not use it.
The use of diatheke—which seemingly does not fit—has given the translators great difficulty trying to determine when Paul meant "covenant" and when he meant "will" or "testament." Why did he even do this when he could have used suntheke? The overall reason is encouraging. Paul wanted to emphasize how much God has done unilaterally—that is, that He took upon Himself to do without consulting with others involved in the covenant—to tip the scales drastically in our favor for the purpose of our keeping the covenant and making it into His Kingdom.
For instance, "God so loved the world that He gave" Jesus Christ in our stead! It was a completely voluntary act on His part. God gives us grace and forgives our sins, and we are justified on the basis of that sacrifice and on the declaration of our faith and repentance. God gives us access to Him in prayer, again based on the work of Jesus Christ. God gives us the very faith that saves. God gives us His Spirit, which is a down-payment of eternal life and empowers us to keep His laws. God gives us gifts, by that same Spirit, to serve Him and the church. He promises never to give us a trial that is too great—which translates into His giving personal attention to each of His children! He promises never to forsake us and to complete the work that He has begun in us.
Some of these unilateral gifts—in a very limited form—appear in the Old Covenant. But it is no wonder that Paul wanted to emphasize better rather than "new." The Old Covenant (because of what God has unilaterally done) is but a pale shadow of the New in terms of what God is working out. It is nothing more than a pale shadow of the promises and the hope derived by those who understand the New Covenant's terms.
To the unconverted who study the Bible, these terms are so enticing that it lures them into concluding that the believer need do nothing. Some will go that far! They will declare that Jesus has done it all for us. They can read the terms, but they reach the wrong conclusion. It leads them to say such things as, "There is no law," and "You don't have to keep the Sabbath. It's just ceremonial." However, the truth is that the covenant is so one-sided, so much to our benefit, that it leaves us without excuse for failure to keep the terms—and those terms include lawkeeping.