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From my understanding they both argue that there is a world independent of our beliefs and perceptions and that this world is causally responsible for what we see but that the features of our experiences – the recognisable objects and their properties – are not features of this causal, independent world, instead being products of the experience.
Could anyone explain how they differ (or if they don't) – or if my understanding of either is mistaken?
Indirect realism in perception is the view that we do not experience the world as how it is, but only through and via our interpretations of how the world is. Representation becomes a key feature of Indirect realism. Consider an object 'a,' and let the image of the object that we perceive be, 'b.' Representation is a resemblance relation, that is, if b resembles a (as how the Representational realist wants it to be), then b resembles a.
Representational realism says that image, b, is by definition visible, however the object, a, by definition invisible. How could the resemblance relation hold between something which is visible, and something which is not? This was alluded to me by my mentor, but the original argument comes from George Berkeley.
Putnam's Internal realism, as I understand it, is the view that the world could be causally dependent on the mind and the divisions we find reality, the categories of being is a product of the mind. In this, it is more similar to Kant's Transcendental Idealism where we experience phenomena but are totally locked out from the noumena. Good thing he abandoned it later on! I think, we can run the same argument we did against representational realism against Kant's transcendental idealism. We could argue that as how reality functions, we always perceive things via resemblance relations that ontologically depend on the noumena, and our perceptions follow from them. But that would be too much to go into, however, I suspect you might have the same suspicion as I did when I first ran into these positions. They are both very similar and have very subtle differences, I would argue that the notion of noumena and phenomena, as how Kant puts it, is just as incoherent as Indirect Realism. However the incoherency is not self-evident and would require me to do more work than the question requires me to. If you want to understand more on how both Indirect Realism and Transcendental Idealism could be false, I suggest Searle's Seeing Things As They Are: A theory of Perception, as well as Peter Coffey's Epistemology Part I and II. The arguments for Direct Realism are complex, so it'll be tough reading anyway.
indirect realism or Putnam's work to comment, but I can definitely grasp a distinction between the two views you mention especially if we see Putnam's view as familial to Kant's. For Kant, it is not that we indirectly perceive things. It is that we perceive things as perceptibles and know them as objects -- operations that are dependent on mind, but when we do so this is how we "know things." And this is the only way we could on the Kantian view.