These are probably two of the most similar churches in Christianity. They share different views on some social issues but besides that, not much.
There are many many differences, between the two bodies. I was raised in the Methodist and Presbyterian church , which are much more similar de to the roots of the denominatons.
Lutheranism was patterned from Reformer Martin Luther a Roman Catholic priest. Although his 95 thesis was posted on the fdoor the Lutheran Church has remained the closest of the Protestant bodies to Catholic in doctrine and liturgy. Although Anglican is also similar to Catholic, there are many points of differences in doctrin which Anglican has more in common with the Reformed faith.
Methodism which came from the word Methodical traces its roots to John Wesley an Anglican priest. The Methodist church became more of an evangelical holliness and revivalist denomiation. Some churches will vary in style. While still having some similarities to anglican, they still retain close resemblance to the Presbyterian church as well. Methodist are much less liturgical than Lutherans and have differences in doctrine regarding the sacraments of communion and baptism.
Lutherans believe in consubstantiation ( similar to Catholics) with regards to Communion while Methodist hold a spiritual significance, much different.
Methodist services can resemble a low episcopal or Presbyterian or Congregational church, but not a Lutheran liturgy at all, which again is much more similar to Roman Catholic.
That depends on the practices of the denomination, but in general the answer is Yes for most main-line denominations (Presbyterians, Lutherans, Methodists, etc.)
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There is no difference.
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The biggest difference among the 3 is style of worship. The Episcopal Church is a bridge between the Roman Catholic and Protestants. It is rich in tradition and rituals. The majority of Episcopal churches are "high church", meaning being formal and adhering to a liturgical practice. The United Methodists are a direct ofshoot of the Episcopal church and shares most of its beliefs and doctrine. Unlike the Episcopal church, the United Methodists have been more willing to adopt contemporary styles of worship, including praise bands and the like. You will find many UMC that are liturgical and are somewhat formal in their worship. The Lutherans are somewhat in the middle of the other 2. All three are sacramental, believe in the Apostolic Succession, and have ordained ministries. All 3 are in a formal communion with each other-allowing for shared preaching and celebrating the sacraments.
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The UK is ostensibly a Christian country divided between Anglicans, Catholics, Methodists and in Scotland Presbyterianism. Judaism and Islam are also strong.
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