I started Paul Hanson's 1975 classic, The Dawn of Apocalyptic. I will start this post by quoting Hanson's definitions of prophetic eschatology and apocalyptic eschatology on pages 11-12:
"Prophetic eschatology we define as a religious perspective which focuses on the prophetic announcement to the nation of the divine plans for Israel and the world which the prophet has witnessed unfolding in the divine council and which he translates into the terms of plain history, real politics, and human instrumentality; that is, the prophet interprets for the king and the people how the plans of the divine council will be effected within the context of their nation's history and the history of the world.
"Apocalyptic eschatology we define as a religious perspective which focuses on the disclosure (usually esoteric in nature) to the elect of the cosmic vision of Yahweh's sovereignty---especially as it relates to his acting to deliver his faithful---which disclosure the visionaries have largely ceased to translate into the terms of plain history, real politics, and human instrumentality due to a pessimistic view of reality growing out of the bleak post-exilic conditions within which those associated with the visionaries found themselves. Those conditions seemed unsuitable to them as a context for the envisioned restoration of Yahweh's people."
Prophetic eschatology is about God working through history and human institutions, as when God uses Assyria or Babylon to punish Israel for her sins, and plans to rebuild Israel through a humanly-administered program that acknowledges her institutions, such as the monarchy and the priesthood. Apocalyptic eschatology, by contrast, desires God's direct intervention into world events as well as despairs about national institutions. The way that this plays out so far in Hanson's book is that there are a group of Mushites (Levites claiming Moses as their ancestor) who remained behind in the land of Israel after Babylon conquered the land. But these Mushites were displaced by Zadokites who were coming back to Israel from the exile, and the Zadokites had the advantage because they were backed by the Persians. The Mushites had no chance of gaining back their power, and so they awaited a dramatic intervention by God himself, in which God would punish the Zadokites and institute a new order. The Mushites attached themselves to the community that surrounded Second Isaiah, and my impression from Hanson's book is that the Mushites invoked the authority of Second Isaiah for their own agenda. The apocalyptic expectation was that God would replace the current system with one in which all of the Israelites were holy, reflecting Second Isaiah's vision of democratization. This vision is contained in Isaiah 60-66, where Hanson sees evidence of marginalization, a Mushite presence, a Palestinian provenance, and a desire for democratization. Ezekiel 40-48, by contrast, upheld the Zadokites.
This is my impression of what Hanson is arguing, and I welcome correction (as long as it's gentle!). Hanson's scenario coincides with other arguments that he makes in his book. Hanson disagrees with scholars who believe that apocalypticism originated as a result of foreign (primarily Persian) influence. He acknowledges that foreign elements were imported into apocalypticism at a later point in time, but he maintains that apocalypticism was descended from prophecy, and that apocalypticism literalized Canaanite myths that were long a part of Israel's heritage (i.e., the Divine Warrior, the divine defeat of chaos, etc.). For Hanson, apocalypticism emerged in post-exilic Israel because a community felt powerless against the establishment in Israel and desired God's vindication and intervention.
This is one take on apocalypticism, and I also checked out another book with an alternative viewpoint, Stephen L. Cook's Prophecy and Apocalypticism: The Postexilic Social Setting. The back cover of this book provocatively states: "Did Israelite Jewish apocalyptic literature originate among alienated or disenfranchised groups? In this overview of apocalypticism in the Hebrew Bible, Stephen L. Cook contends that such thinking and writing stems from priestly groups that held power." So Cook believes that apocalyptic was an establishment phenomenon, not a product of disenfranchised groups. I'll read Cook's argument, however, after I finish Hanson's book.