In Romans 14 Paul is trying to guide congregations to preserve the unity of the body of Christ despite having a wide variety of views on some practices. He mentions two in particular, considering some days sacred and eating some types of foods.
Esteeming one day over another may refer to when the Roman congregations chose to gather. The natural assumption is Jewish Christian congregations continued to worship on the Sabbath. Primarily Gentile congregations met whenever they could, apparently settling on Sunday, the day Jesus rose from the dead.
Eating and abstaining may refer to Jewish food taboos. Again, when a primarily Jewish congregation shared a meal, the food would have been purchased and prepared with attention to cleanliness (i.e., not meat sacrificed to idols, nothing forbidden in Leviticus), etc. Primarily Gentile congregations may not have adopted Jewish food laws, accepting all foods as clean after one gives things for the Lord for the food. However, it is likely some Gentiles did choose to avoid food sacrificed to idols.
What matters for Paul is living one’s life “for the Lord” and not for ourselves. This means the one who is in Christ (a living sacrifice, one who is living in a way that promotes unity in the body of Christ), ought to voluntarily set aside preferences in deference to others.
Voluntarily setting something aside is the key to understanding the principle Paul wants to establish here. Like Jesus, who set aside certain rights he had as a member of the Godhead in order to become human (Phil 2:5-6), so to the member of the body of Christ in the present age must set aside their privileges the may legitimately be owed in order to preserve the unity of the Body of Christ.
Paul is not discussion sinful practices, but what are often called preferences. He is not talking about Gentiles visiting a prostitute (as he is in 1 Corinthians 6), since that is a practice incompatible with being a living sacrifice, holy and acceptable to God. This is the nature of the strong/weak in this passage: the person with weak faith considers eating food to be a mark of spirituality and therefore breaking those convictions would be a sin.
Does this only work one direction? A person who does not eat unclean food cannot “give up” their preferences and eat unclean food to make a Gentile feel comfortable? For example, if a person today is a vegetarian, can they “give up their conviction” and share meat with someone who eats meat? If I were to share a meal with a Seventh-Day Adventist, for example, I would have no problem eating any food they served. But they may have a serious problem eating something I serve. If I have a meal in an Israeli hotel, it is far easier for me to eat kosher than to insist on my rights and have the kitchen make me a bacon-wrapped cheeseburger.
It is far easier for the meat-eater to give up their conviction and eat only vegetables. This is certainly true on a physical level. But more importantly, with respect to convictions, the meat-eater is not violating a principle of their faith, but the vegetarian would be “sinning” with respect to their own world view.
There is a clear application of this principle for the modern church. I think there are some easy examples: If a member of congregation prefers one style of music for worship, they ought to be able to set that preference aside in order to reach people for Jesus Christ.
But I can imagine other situations which would make some Christians more uncomfortable. Could a pastor drink a beer with someone in order to not make a beer drinking member of their congregation comfortable? What about a pastor trying to reach a person in the south who is offered a wad of chewing tobacco. Could they accept the offer without violating their conscience? It is critically important to observe Paul is talking about practices which are not important for salvation in the present age nor is he talking about sinful practices (even if the weaker brother thinks they might be).
Both the weak and the strong are believers, and both are welcome in Christian worship and fellowship. For Paul, these are not matters to divide churches or break fellowship over. What are some problems you have encountered trying to find the right balance between preferences in local congregations?