The contrast of these two characters, Edgar and Heathcliff, really exemplifies a common conflict in literature, that of forbidden love. Simply put, Edgar's love for Catherine provides her with safety, security, and a comfortable lifestyle; he can even be credited, perhaps, with having turned her into a lady during her stay at his home. Heathcliff loves her in a way that borders on the obsessive, and he isn't really capable of the empathy, commitment, and/or sacrifice required to make a long term relationship, i.e. marriage, work. However, this very quality in him seems to draw Catherine to him, as well as the romantic notion that they were/are/will always be soulmates. Catherine didn't seem to have much interest in the notion that they were soulmates when she first returned from her stay at the Lintons', but when Heathcliff returns after a three year absence as a handsome and strapping man, she seems to have an epiphany of sorts.
The conflict in this novel is not unlike the conflict found in Edith Wharton's The Age of Innocence, where Newland Archer marries one woman for reputation, appearances, and social stability, while he secretly adores another. It is also similar to the conflict found in the pop culture phenomenon of the Twilight vampire saga by Stephenie Meyer; Bella struggles throughout much of the series with her feelings for Jacob, which tend to be platonic in nature, versus her passion-to-the-point-of-obsession for Edward.
Heathcliff and Edgar have this good angel bad angel complex going on. Heathcliff would be deemed the bad angel in this case. You can tell by the way he characterized as always being reclusive, dark and brooding. Heathcliff's hair is also black which you could say is black halo. Edgar on the other hand would be good angel. Edgar is characterized as a well mannered boy who has little to no temper. Edgar's hair is blond so I guess you can say it's the color of his white halo. Also Heathcliff is painted as a villain and evil while Edgar is seen as a loving caring father.
Now they both are same when it comes to loving Catherine. They both love her some much that they are willing to do basically anything for her even if it means standing their position of what claim they have in her life. That being Heathcliff with him being stagnant on his being the love of Catherine's life and Edgar on his being Catherine's loyal, unwavering, dominate husband.