If we were in any way to compare (and not contrast) Homer Barron with Mr. Grierson there may be one thing in common: They both represent Emily's every weaknesses and fears.
Emily cannot let go of neither nor because they both represent security, company, solidarity, and in some wicked way- love.
They are also the shields that Emily has to defend herself from a world of which she is completely unprepared.
Therefore, the men are hard to compare in terms of their characters' similarities, but they could be connected in that Emily needed both of them to survive. They, therefore, are both meaninfgul symbols of safety, security, and love in Emilys life.
In Faulkner's gothic story, "A Rose for Emily," while the differences between Mr. Grierson and Homer Barron are distinct as outlined above, there is one similarity between Emily's Southern aristocratic father and Emily's working-class Northern beau. This is stated by the townspeople after Emily denies that her father is dead and the authorities must persuade her to let them dispose of the body, and, as they are about to resort to law and force, she breaks down. The narrator remarks,
We did not say she was crazy then. We believed she had to do that. We remembered all the young men her father had driven away, and we knew that with nothing left, she would have to cling to that which had robbed her as people will.
Just as her father's body is "clung to" by Emily so, too, is Homer's body. For, Homer, like her father, has robbed her of respect and of love and of her feminine desires.
It would be difficult to find similarities between the two men, other than their involvement in Emily's life, but the differences are sharp and significant. Emily's father was selfish and domineering, a product of the genteel Old South, bound by its traditions and social roles. As head of the exalted Grierson family, he demanded complete devotion and obedience from his daughter, not allowing her to pursue a life of her own. When Emily was a young woman, her father drove away her suitors, keeping her at home and under his control, for his own comfort and satisfaction.
Homer Barron, in contrast, was a Yankee and a construction worker, a strong, attractive, masculine figure who came into Emily's life quite unexpectedly. Homer had no regard for the finer points of the Southern class system, flaunting his affair with Emily in a very public manner. As they drove through the streets together in an open carriage, their behavior was considered scandalous by the citizens of Jefferson. Homer represented all that the gentle people of Jefferson deplored, and he was above all, a Yankee. In pursuing her relationship with Homer, Emily was repudiating her father's values, his control, and her own social position in Jefferson--another controlling force in her life.