What are the differences between a schema and a mental model? Do they overlap, or is the mental model a subtype of the schema?
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.
Schemata are said to account for generic concepts and the meaning of those concepts. Schemata are generally said to represent ``knowledge'' however, it is argued that the theory struggles to account for specific knowledge. Most theorists seem to agree schemata are meant to represent generic concepts - not specific facts or information. I would recommend "Schemata: the building blocks of cognition" by Rumelhart if you want to read more about this. Category theory might also be interesting to look into just to get an idea about the types of things schemata represent.
Mental Models are structures which are said to represent some aspect of the environment. Where schemata represent concepts, mental models represent an understanding of the specific actual environment a person is in/interacts with. I would recommend Normans "Some Observations on Mental Models" for more information.
One example I like is the following: You can have the concept of a house somewhere in memory. When you are shown a photo of any house or go into a house, even if you haven't seen it before, you still recognise it as a house. You can recognise this because you have one or more schemata which have encoded the ``concept'' of what a house is.
However, if you were living in a house, you might know the bathroom is two doors down the corridor to the right, or that the kitchen tap needs to be turned on in a particular way to get hot water, or that the second step creaks in cold weather. This information doesn't really relate to schemata - this is specific information about the environment you're in. This type of information is stored as a mental model of your specific house.