Those on both sides of the arguments for remaining in the UMC and disaffiliating from the UMC are asking the question, "What difference does denominational affiliation mean?"
It's an important question and one that everyone should seriously consider.
What difference does it make? What's it matter what sign is on the church? What's in a name?
For some folks, the name might not matter. You might attend the church that is closest to your home, the one you grew up in, or the one where your family or friends worship, regardless of its denomination or how its beliefs align with your own. While I think there are some who fall in to this category, I think it's a slim minority.
For most people, though. the name matters. Denominational affiliation is more than a 'brand,' it is a confession of faith. It lets you know where a church stands on lots of beliefs and issues before you even walk in the door.
It lets people know what the church believes and teaches. Walking in to a Roman Catholic church, you can expect to be taught certain things about Mary, the Eucharist, and what happens after we die (Heaven, Hell, and Purgatory). By the same token, if you walk into a Presbyterian church, you know you will be taught certain views about predestination and election, much different than what you'd hear in a Roman Catholic parish. You can also learn a lot about where a church stands on societal and moral issues (drinking, gambling, marriage/divorce, abortion, euthanasia, etc.) because most denominations have statements on issues like these and require their clergy to teach and preach in accordance with them.
It says something about how a church is going to worship. If you walk into an Anglican church, you should expect to worship according to the liturgy and services in the Book of Common Prayer because this is mandated by the various Anglican Communions. By contrast, if you head over to the local Pentecostal church, you ought to expect worship to look very different from an Anglican church and probably even a little different from a neighboring Pentecostal church. Though not as obvious as preaching, how a church worships says a lot about what it believes.
It says something about how a church views itself as part of the larger Christian church. Orthodox, Roman Catholic, Anglican, and Lutheran churches typically feel very strongly about maintaining continuity with 2,000 years of Christian teaching and practice. While basing their beliefs on Scripture, they may also appeal to important figures in church history, like the Church Fathers or important Reformers. Baptist and Pentecostal churches will usually be much less concerned about the ancient traditions and teachers of the church, focusing more on important preachers and teachers of the last couple hundred years. Non-denominational churches, by their lack of affiliation with a denomination, stand alone in the "crowd" of churches, in theory not connected to any Christian tradition (though almost always baptistic in theology and contemporary or even charismatic in their worship) and not connected to any other group in their work, mission, or life.
(There are, of course, lots of other things tied into a denomination: how pastors are trained, how pastors are called to a congregation, how the hierarchy of the church functions, etc. While these are important matters of polity, they should not have significant impacts on the theology proclaimed and lived by a local church.)
With that in mind, there are several things that should not be affected by denominational membership. These are also important because there are things all churches should be doing, regardless of their name.
It should not say anything about whether a church is loving. All churches, regardless of their confession or denomination should welcome sinners, desire their salvation, encourage them, and come alongside them in the struggles of real life to grow in grace and the likeness of Christ. This is one of the principle works of Christ's church on earth.
It should not say anything about whether a church participates in mission and charity work. Again, all churches, should endeavor to feed the hungry, shelter the homeless, aid the poor, minister to those in prison, and care for all. This is another non-negotiable for a faithful Christian church.
It should not say anything about whether a church has a positive and lasting impact on their community. Christians are called to reach out, engage, and transform the world for Christ. Exactly how this looks varies from congregation to congregation, even in the same denomination, but the mandate is the same for all. A church that is turned in on itself and disengaged from the community around them isn't a church in any Christian sense. It's either living in disobedience to Christ or it's a cult.
Does denominational affiliation matter? Absolutely. It may not matter in all the ways you think, and there are things that it shouldn't affect at all, but church names are ultimately confessions of faith and mean a lot.