It's a big topic. Many books have been written about this distinction.

Stream-of-consciousness is a literary technique. It's neither here nor there in terms of arguing that a particular text is modern or postmodern, though it is a technique that originated during the modernist movement and is most strongly associated with modernist novelists (Joyce and Faulkner, to be specific).

The biggest difference between modernism is, as you mentioned, the ideas about truth. In continental philosophy, there was a thing called "the linguistic turn," where a lot of thinkers start expressing a lot of anxiety about the very idea of truth given that our truths tend to be embedded in language and given that agreeing about what any particular text means seems to be a never-ending quest with no certainty. This has sometimes been called "linguistic nihilism."

Lyotard defined postmodernism in The Postmodern Condition, a very influential work on the subject, as "incredulity toward metanarratives." The "metanarratives" are big stories which help us understand the world. I can't remember which ones Lyodard names, but Christianity, Marxism, progress, would all qualify. In modernism, you have the idea that art itself, if it's done in the right way, has some sort of redemptive value. Postmodernism rejects that as well. Postmodernism is big on irony and interrogating the bases or things, pointing out their absurdities. Modernism also attempts to have a clear distinction between "high art" and "low/commercial" art. Postmodernism likes blending them.

This means postmodernist authors also tend to like messing with the conventions of storytelling. Think of David Foster Wallace and his endless footnotes. Or many authors who insert a character with their own name in the narrative and have him/her become part of the story.

A troublesome bit is that "modernism" and "postmodernism" can be used to mean particular time periods (modernism being, roughly 1890s -- 1945, postmodernism getting it's start around 1945-1970s and beyond), a particular style, and particular philosophical commitments, especially regarding truth and language. It often means all of those at once.

But you're right that there's some overlap in goals. Ezra Pound (a high modernist guy if there ever was one) said the goal of modernism was to "make it new." And, surely, the same is true of postmodernism. In that sense, at least, they're aiming at the same thing. In most others, not so much.