This book really is about unrealized hopes and dreams, isn't it? The most prominent of these are Gatsby's hopes and dreams, to achieve a social status that he was not born with, to achieve wealth, and to win Daisy, whom he has loved since they met. The entire book is about his longings, as he buys a mansion, acquires numerous material possessions, gazes at Daisy's house across the water, and holds magnificent parties in the hope of attracting her to his house.
But these hopes and dreams of Gatsby's are a symbol, too, of the hopes and dreams of those who have come to America. Nick expresses this on the very last page of the novel, when he says,
...I became aware of the old island here that flowered once for Dutch sailors' eyes....Its vanished trees, the trees that ad made way for Gatsby's house, had once pandered in whispers to the last and greatest of all human dreams; for a transitory enchanted moment man must have held his breath in the presence of this continent, ...face to face for the last time in history with something commensurate to his capacity to wonder (189).
Nick goes on to say the dream "eluded us then" (189), but it does not matter because we will continue to pursue this dream. Gatsby is a symbol of the American dream of a fresh start, of an ability to leave one's lowly beginnings behind, to invent oneself in a New World, and to woo and win an unattainable princess.
Fitzgerald uses the words especially in regards to Gatsby and early American settlers. He does so because the novel's main moral is that the American dream is unattainable. Its followers are constantly hoping for its fulfillment, and just like Daisy's green light is to Gatsby, the dream is always just beyond reach.
This idea--that the American dream no longer exists for all who are willing to pursue it--is the basis for the literary era (Modernism) with which Fitzgerald is connected. Modern American writers express their and most of the country's disillusionment with American ideals in general.
So, because Gatsby is a treatise on the cruelty of the tantalizing American dream, Fitzgerald uses "hope" and "dream" to illustrate to his readers that even those who cling faithfully to those two elements end up empty handed.