How do authors take similar ideas and themes and present them in different ways?
In brief, writers will opt to examine ideas according to genre demands (some writers will use poetry and others fiction or drama and these choices will determine to some extent how an idea will be explored), according to stylistic preferences (some will use complex prose and stream-of-consciousness like Virginia Woolf while others will use "hard-boiled" prose like Ernest Hemingway), and according to different conceptions as to what kinds of characters and stories might best facilitate a particular theme or idea.
This question might best be answered by looking at some examples. You might be familiar with The Great Gatsby and Death of a Salesman. Fitzgerald's novel (Gatsby) is often understood as being interested with the idea of the American Dream.
"Whether one calls it extravagant expectation, faith in entrepreneurial success, or confidence in the American cornucopia, the American Dream is an inherent part of social, cultural, political, and literary America." (eNotes)
This thematic idea is rather broad, but can be connected to notions of social-climbing, financial mobility and the achievement of wealth. In short, the American dream can be said to be defined by the idea of moving up through self-determination. At least, this is how Fitzgerald's novel treats the American Dream.
Gatsby examines the failings of the American Dream as it seems to lead to a loss of innocence.
"The American Dream rings hollow as wealth results in not contentment, but corruption and moral dissolution." (eNotes)
Fitzgerald uses a relatively complex narrative structure with a narrating character who is not the protagonist to tell a tale of social striving and moral failure that both champions the innocent and romantic vision of self-determination and decries the tendency that this pursuit has to leave casualties in its wake.
In contrast, Arthur Miller's play (Salesman) deals with the American Dream as a burden that has insinuated itself into the expectations of generations of Americans leading to an empty materialism defined by outward and socially-oriented success. Miller's critique on the American Dream is in some ways similar to that presented by Fitzgerald in Gatsby, however the theme is explored in very different ways.
Miller tells the story of a family that is falling apart. The system of patriarchy is no longer functioning because the father figure has lost the respect of his son, lost his self-respect, and cannot seem to fix on a way to regain any of the things he has lost. Instead, the protagonist succumbs to a dementia defined largely by the trappings of the American Dream.
"Willy constantly strives to be financially and socially successful but fails to recognize that his ideas of success do not fit with social expectations." (eNotes)
He imagines that he is speaking with his brother, an adventurous and extremely wealthy self-made man. He imagines also that he is well-liked up and down the east coast. He retreats into this imaginative setting where ideas of social achievements are paramount. In doing so, Willy Loman refuses to address his actual achievements and seems to be running away from his actual failures as a husband, as a father and as a salesman.
In Gatsby, the dream carried by Jay Gatsby is the only redeeming quality of the character. In Salesman, the dream carried by Willy Loman is the source of his anguish and chagrin.
Thus these two texts examine the American Dream and bring criticisms to it, but do so from different approaches. Not only is this true in how they treat the symbolism of the dream, but it is also true in the differing narrative structures employed, the character relationships centrally used, and in the genres of the texts. One story is told as a novel and the other as a play.
To back up to an overview of this discussion, we should note that the ideas at work in these texts are similar but not necessarily the same. Writers will take up the same concepts like the American Dream and draw connections to other ideas (like moral obligation versus self-interest or the dissolution of a family) and offer different conclusions or thoughts on these concepts. The central idea is the same, but the emphasis is different.
Also, the genre that one writer might work in may differ from that of another writer. Poets like Maya Angelou have written about race issues in America and so have novelists like Richard Wright. Their techniques are necessarily different due to genre constraints even if their broadly-defined subject is the same.
Additionally, we have noted already that a writer's choice to locate an issue within a certain narrative structure and to attach the issue to a certain set of characters has a place in this conversation. In The Adventures of Huckleberry Finn, Mark Twain set Huckleberry Finn out on a journey to explore biases and prejudices of all kinds and learn to think for himself. This choice and the setting on a river along with the linear narrative structure becomes intimately bound to the underlying themes and active symbolism of the text.
Compare Twain's approach to prejudice as a theme to Faulkner's approach in Light in August and we see two very different novels on the same essential theme. Faulkner uses a small cast of characters, each of which are attached to a different kind of bias (being poor or ethnic or publicly disgraced) and spins a tale through a complex interweaving of story-lines and flashbacks.
Both Twain and Faulkner suggest that prejudice is a complex issue that exists both internally (absorbed into the psyche) and externally (socially), but the structures of their works and the characters attached to the themes serve to frame the concept of prejudice so that we see the idea differently in each work. So even within the same genre, writers will explore similar themes and topics, often coming to similar conclusions, but approach these ideas through a variety of characters, narrative structures and stylistic modes.