How do the Personal Ordinariates differ from Anglicanism? In 2009 Pope Benedict published an Apostolic Constitution called Anglicanorum Coetibus providing for the creation of Personal Ordinariates  for Anglicans who wished to join the Roman Catholic Church.

This, it says, was to maintain the liturgical, spiritual and pastoral tradition of the Anglican Communion within the Catholic Church as a precious gift ... a treasure to be shared.

As far as I know there are now three of these Personal Ordinariates, one for Great Britain, one for North America and one for Australia and Japan.

In 1906 when some members of the Russian Orthodox Church formed an Eastern Rite 
church, the Russian Greek Catholic Church, and asked Pope Pius IX what they needed to change in their Rite he said "nec plus, nec minus, nec altera" -  add nothing, subtract nothing, change nothing. 

I understand that a different approach was adopted with the Anglican (ex-Angllican?) Ordinarates than with the Eastern Rite Churches, but how does the doctrine or practice differ in the Ordinariates from in the Church of England (for example)? What has been added, what has been taken away and what has been changed?  

(I have looked at the Ordinariate website and can't find anything specific.)
In terms of what has been added, subtracted or changed: either “not a great deal” or “a huge amount”.
What has happened is that members of the Anglican Church have been received into the Catholic Church and become Catholic. That means that they have subscribed to the entirety of the doctrine of the Church, including the Real Presence and other things that Protestants might find difficult such as the dogmata concerning the Virgin Mary.
In most cases, those who left the Church of England would have believed all those things anyway, so not much has changed pragmatically. In all cases, they are believing them in a Catholic context, so a huge amount has changed: not least that if the Eucharist is the Real Presence now, what was it then?
This is the principal reason why “Anglican Ordinariate” is an egregious misnomer. We are not Anglicans, and the Ordinariate is not an extension of the Anglican Church; it’s not an exclave of Anglicanism within the Catholic Church. We are Catholic, and the Ordinariates are Catholic.
In terms of externals, because the Ordinariates were formed from the “high” end of the Church of England, on the whole, you will generally find in Ordinariate worship compared to diocesan Catholics a higher level of ceremonial and greater use of music (and more participation in congregational singing). The Ordinariates’ founding documents stipulate a greater lay involvement in governance than in diocesan parishes. The Anglican ethos of caring for the soul of everyone in a territorial parish seems to be more to the fore than in diocesan parishes, where the focus is more [but certainly not exclusively] on the congregation. These are all the “treasures to be shared” that the founding documents mention.
To turn to specifics, the Ordinariates now have their own Missal as part of a planned series of worship books called Divine Worship; the Missal bears the title Divine Worship: The Missal*. It’s very similar to the 1662 Book of Common Prayer and the American prayer books, particularly those of the Pastoral Provision. Needless to say, it doesn’t contain the Protestant material introduced by Cranmer — but the Mass does have a large number of Cranmerisms, including the confession. In my case, we use it three times a week and even sing Merbecke on Sundays.
The Ordinariates were enjoined to be distinctive but not isolated; to integrate without becoming assimilated. Drawing on the “distinctive patrimony” was one method by which that could be achieved, and there was debate about what patrimony actually meant, especially in practice. The general conclusion is that it means heritage, and there is an enormous amount of heritage on which to draw, especially as what has come to us via the Reformation draws on earlier liturgy and practice. That pre-Reformation heritage is now available to the Catholic Church in three forms: the strand that has come through the Second Vatican Council; the “Extraordinary” strand which came through the counter-Reformation of the Council of Trent and culminated in the Missal of 1962; and the strand which preserved certain aspects differently, within the Anglican Church. What the Anglican Church rejected has often been restored; what it added to express its protestantism has been removed.
I wrote a blog post in 2016.

Needless to say, Divine Worship: The Missal is the treasury at the centre of the worship and liturgy in the Ordinariate. It unites us with the Church of the earliest English saints and it draws on the work of past liturgists in which the Holy Spirit can be discerned. The language, the structure and the options available show the heritage of our Mass and, despite what people may think are its roots, how it is firmly set in the Western Rite of the Church, just as the English Uses of Sarum, York and the others were before it.
Perhaps we are fortunate that the Reformation liturgists faithfully translated a great deal of the Latin they inherited. While the defective form of the Communion service had to be corrected — and this was attempted even within the Church of England well before Series Two in 1965 — the resonant words they created have served Christians in this land and others well for 450 years, and continue to do so.
But it is absolutely necessary not to wallow in the past! Many of the Ordinariate may remember Series Two, Merbecke and Shaw, and many diocesan Catholics will remember the Interim Rite, but this isn’t an exercise in comfortable nostalgia. Every celebration of the Eucharist is a re-creation of the Sacrifice of the Mass: Christ’s One Sacrifice united through space, time and eternity for all believers. Although modern language is certainly easily understood and relevant to today, using language which is out of time can help to show the eternal too. It is the calling of the Church to teach the unchanging Faith anew to every generation, and the Use we have been given for the celebration of the Mass is — perhaps counter-intuitively — a Fresh Expression of worship.


* That link shows the small edition suitable for the bookshelf in the study. There is a huge altar edition as well and a middle-size one too. All are identical in content, including the colour illustrations. There’s a People’s Edition which has the propers and readings and Martin Travers’ illustrations from the English Missal.