There is, indeed, a similarity in theme and style in William Faulkner's short stories "Barn Burning" and "A Rose for Emily". The basic similarity is that both stories explore the extent to which maintaining ties with the past actually helps us move into the future.
Thematically, both stories depict life in the American South, along with the idiosyncrasies, expectations, and dynamics that take place in traditionally small, bucolic settings. Faulkner tells the stories of Emily and Sarti considering the perspective of the modern reader and how the current views of the world affect the interpretation of life in the Old South.
In "Barn Burning" a perennial theme is family versus society. We see a very enmeshed and dysfunctional family led by a man with no morals inviting his sons to conduct criminal behavior. The ties that bond Sarti to his father make him feel obligated to comply with his father's wishes but, in a moment of maturity and self-indentity, he breaks his ties with a legacy that would have brought about his own ruin.
In "A Rose for Emily" the strong ties among Southerners are explored differently as Col. Sartoris (notice the name-swapping in both stories) makes exceptions for Miss Emily due to her lineage and family legacy in Jefferson. However, as time changes, so does the treatment given to Emily, which she strongly resents.
Moreover, Emily is also bound to the wishes of her father: A man who also overpowers her free-will, and renders her unable to deal with life's everyday realities. Emily, in contrast to Sarti, does not become independent from her father. Rather, he dies leaving her devastated and even weaker. Still, she holds true to the need to honor the dignity of her family's Old South roots.
The ties of the Old South are questioned in both stories: Is it worth sticking to an old bond when it is obsolete? Is there an actual obligation to honor the past?
Hence, although there is a detour in theme in terms of the individual experiences of each character, the overall theme of tradition versus independent thinking permeate both stories in a great way.