Differences between the two dudes:
Minsu: Boyfriend out of circumstance; after years of dealing with dudes confessing with ill intentions, she's found someone who isn't clearly assertive, and finds comfort in it, so tries going out with him. Looks to him to make for herself an "ideal" attitude. We've yet to see Minsu do anything like what Jaewon has done.
Jaewon: Someone who warmed up to her (and vice versa) after a bad first impression. Also similar in personality and experience; someone who is "similar" to herself. Sure, he says he doesn't give much of a shit about her, and yet goes out of his way to defend her - much like how she helps the manager and Jaewon get it on.
Point is: Jaewon actively gives more of a shit than Minsu; so she's "probably" going to break up with Minsu. (But like Loki said, cover art. Obviously they're breaking up lol)