What attracts Eveline more than anything else is the prospect of escape from a life of drudgery, hardship and suffering. The powerful memory she has of how her mother lived her life and how she ended her days is the fate that Eveline knows she will be able to avoid if she leaves Ireland. Note how Eveline characterises the life of her mother as she contemplates her situation:
As she mused the pitiful vision of her mother's life laid its spell on the very quick of her being--that life of commonplace sacrifices closing in final craziness.
Her response as she contemplates her mother is to think: "Escape! She must escape!" This is what attracts Eveline to running away, even though, interestingly, as she finds herself in the position of just being about to leave, she seems powerfully drawn to stay, as shown by the passivity of her body as the story opens, as she leans her head against the glass of the window and looks down on the street where she lives.
As for Jimmy in "After the Race," it is clear that his motivations are very different. As a boy of an Irishman who has made it rich, it is clear he has been spared no luxury to help him on life's way. The only problem is what Jimmy clearly desires is wealth and status. Jimmy is attracted to his friend, Segouin, because he "had the unmistakable air of wealth," and Jimmy desires nothing else but to be able to keep up the appearance of wealth, even when this involves foolishly losing vast quantities of money that his father has worked so hard to earn. In the gambling that ends the story, Jimmy recognises that he will lose and that the responsibility for this lies with him:
But it was his own fault for he frequently mistook his cards and the other men had to calculate his I.O.U.'s for him.
Jimmy is a man who seems to be trapped in a world of endless consumption and wealth where he suspects he doesn't really belong but from which he is powerless to break free. Both Eveline and Jimmy are therefore very different in terms of what attracts them and their reasons for that attraction.