The one who cares most is the one who has more interest in defending the relationship’s existence. Ergo, they’ll be more likely to comply with the partner’s demands, if they feel the relationship is in danger.
As an example, my girlfriend used to smoke, a lot, like a half package on a good day, whole package on a bad day, usually something in between. My father is also a smoker, and last year he almost died of pulmonitis because of it. We all warned him about smoking 10–15 cigarettes a day when underweight and pushing 60, but his only reply was that he liked it so he wasn’t going to stop.
Way before my father’s close encounter with death, i asked my girlfriend to stop smoking, because I found my father’s behaviour incredibly irresponsible and shortsighted for someone who decided to have a family of his own and therefore also has a modicum of responsibility towards somoene else, and i didn’t want a relationship with someone who would ultimately choose personal pleasure before committment to their partner/children/etcetera. In that incident i cared more about the issue than the relationship.
In this case you can have 2 outcomes:
the other person cares more about their side of the issue as well => breakup
or
the other person cares more about the relationship => attempt to comply
Note that i say “attempt”, my GF tried, back then, but it took her 3 tries and seeing my father in the hospital with a pulmunary drainage pump after being drained almost 3 litres of fluids and risking a heart attack to make it permanent.
So yeah, in the event that one of the two people in a relationship is unhappy with something, if they care less about the relationship than the issue (which isn’t to say they don’t care about the relationship, we’re talking relative terms here) they have the power to force the other person to decide what comes of the relationship through word or action depending on the case.