I'm going to assume that you are referring to American fiction, but if you are not, most of the ideas below will still fit.
Modernist and Postmodernist Fiction both stress disillusionment with traditions and ideals. In American Literature, this means that authors such as Steinbeck and Fitzgerald portray the American Dream as unattainable and as a cruel illusion.
Another similarity is that American Fiction became more multi-cultural during both literary eras. During the Modernist Movement in America, the Harlem Renaissance occurred, and the rhythm of jazz and blues made its way into poetry. In comparison, the Postmodernist Era has produced American writers from a myriad of ethnic backgrounds. These writers, just like those from the Harlem Renaissance, have added richness to the culture of American Lit.
One of the most significant differences between the Modernists and Postmodernists is the theme of conformity versus individualism. The Modernists wrote for readers who had been through several traumatic periods (World War I, the Stock Market Crash, and the Great Depression). Those readers seemed to long for a sense of normalcy. For some, that normalcy came from Europe where so many of them had been during the war. So, while Hemingway often wrote of far-away places, he still dealt with intensely American issues. Likewise, poets such as Ezra Pound and T.S. Eliot introduced Imagism and Symbolism which possessed the flowery style of past eras.
In contrast, the Postmodernists found themselves living in a nuclear, space-traveling world. Because of the rise of technology, Postmodernist fiction often features people struggling to establish their individualism rather than conforming. The Beats and other avante garde poetry strayed from the Modernist style. Poetry became much more personal during the Postmodernist era through the influence of poets like Robert Lowell.
The most significant difference between the two literary eras is the rise of nonfiction's popularity under Postmodernism. I've seen the influence of this change simply by teaching high school English. More and more tests focus on students' ability to analyze nonfiction, and nonfiction bestsellers have become just as interesting and popular as bestselling novels.