One interesting difference is how the word worship (proskuneo) is really only used in a positive sense in the Gospels and in the Book of Revelation. In these two literary types, the presence of Christ is localized and thus bowing down in worship is appropriate. However, the word is only used something like six times between the Book of Acts and all of Paul’s writings; in these letters, the word is never used in a positive sense. Moreover, it is never used in the epistles of Peter, James, or John. Since we are the temple of God, all of life is now a spiritual act or service (latrao) of worship.
Another interesting aspect of the difference between the Old and the New is the extent to which worship is prescribed. In the OT, worship is prescribed down to the very color and type of thread to be used in each ceremony. In the NT, we have almost no explicit prescription for how our worship service is to transpire.
Since Piper knew that some of these observations would be disconcerting to many Calvinists, Piper quotes Calvin to show how much liberty we have under the New Covenant for how we worship. As you read this, it is helpful to think about how the Regulative Principle has been overworked in some Reformed circles.
But because [God] did not will in outward discipline and ceremonies to prescribe in detail what we ought to do (because he foresaw that this depended upon the state of the times, and he did not deem one form suitable for all ages), here we must take refuge in those general rules which he has given, that whatever the necessity of the church will require for order and decorum should be tested against these. Lastly, because he has taught nothing specifically, and because these things are not necessary to salvation, and for the upbuilding of the church ought to be variously accommodated to the customs of each nation and age, it will be fitting (as the advantage of the church will require) to change and abrogate traditional practices and to establish new ones.
(Calvin: Institutes of the Christian Religion, 4.10.30)
As always, this is why I love Calvin. When I read his writings I find that he is much more reasonable than many of his students and detractors make him out to be.