What are the differences between a schema and a mental model? Do they overlap, or is the mental model a subtype of the schema?
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At a high level, schemas and mental models are of a kind in cognitive science. You can think of both as cognitive or representational structures that aid in the storage and retrieval of information. Schemas were introduces earlier (Bartlett, 1932), and were followed by concepts such as scripts and frames (Schank & Abelson, 1975; Rumelhart, 1972). All of these, in general, describe the way that information can be structured in the mind such that we can easily navigate complex yet familiar situations, as well as adapt to and learn how to navigate novel situations.
I usually attribute the introduction of mental models to Johnson-Laird (1980, 1983). A lot of this work was based on evidence of spatial representational structures in peoples' minds when they do things like read a story, and (since I'm a language guy), these ideas were further extended to situation models of text, which suggest we hold mental representations outside of the realm of space and time, and into causality and goals (e.g., Kintsch & van Dijk, 1983; Zwaan & Radvansky, 1998; and others such as Glenberg, Morrow, ...). I kind of think of this work as extending more abstract notions such as schemas into people's actual memories and experiences.