* Relations strained recently, especially over women priests
* New leaders seen as more worldly-wise than predecessors
VATICAN CITY, June 14 (Reuters) - Pope Francis and the new head of the world's Anglicans acknowledged deep differences over issues ranging from gay rights to women priests but pledged to seek unity when they met on Friday for the first time since both took office in March.
Relations between the Catholic and Anglican churches have been strained for years, especially over Anglican ordination of women as priests, and the meeting at the Vatican was billed as an opportunity to reduce tensions.
Welcoming Archbishop of Canterbury Justin Welby to the Vatican, Francis called for Christians to work together to protect the "foundations of society" such as respect for human life and the institution of the family built on marriage.
Francis was inaugurated as the leader of the world's 1.2 billion Catholics on March 19, following Benedict's abdication, and just two days before Welby officially took over from Rowan Williams as head of the 80-million-strong Anglican Communion.
Welby said on Friday he hoped the proximity of the two leaders' inaugurations would "serve the reconciliation of the world and the Church", while noting the difficulties ahead.
"The journey is testing and we cannot be unaware that differences exist about how we bring the Christian faith to bear on the challenges thrown up by modern society," he said.
Anglican ordination of women is a thorny issue between the two Churches, with the Vatican firmly opposed to female priests, and attempts by Francis' predecessor Benedict to woo disaffected Anglicans back to Catholicism has caused more friction.
In 2009, Benedict decreed that Anglicans who feel their Church had become too liberal could find a home in Catholicism in a parallel hierarchy that allows them to keep some of their traditions, such as parts of the Anglican liturgy and the Anglican Book of Common Prayer.
16TH CENTURY SCHISM
It was the boldest step by the Vatican to welcome back Anglicans since King Henry VIII broke with Rome and set himself up at the head of the new Church of England in 1534.
Francis said on Friday he was sure the move would help the Catholic world to better appreciate and understand the spiritual and pastoral traditions of the Anglican community.
Welby, a former oil industry executive, has inherited a Church which is itself divided over issues such as gay rights and women bishops. He is against gay marriage but favours female ordination, and is stuck in the crossfire between liberal and conservative clerics.
In January this year, the Church of England lifted a ban on gay male clergy who live with their partners from becoming bishops on condition they pledge to stay celibate, deepening a rift in the Anglican community over homosexuality.
The Church, struggling to remain relevant in modern Britain despite falling numbers of believers, published a plan in May to approve the ordination of women bishops by 2015, after the reform narrowly failed to pass last November.
Welby, a pragmatist hardened by years of work as a crisis negotiator in Africa, is seen as more down-to-earth than his academic predecessor. In that way he presents a similar image to Francis, who led an ordinary life close to the poor as Archbishop of Buenos Aires.
On Friday both men stressed the need for Christians to help poor people and promote social justice and peace.
After addressing the pope and Vatican officials, Welby went on to pray at the tomb of Saint Peter under St. Peter's Basilica. (Editing by Mark Heinrich)