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What is the difference between sparsity and sparseness? Is it that the latter emphasises the extent to which something is sparse? I'm not sure I can see the thin line that separates them.

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I think you will be hard-pressed to find any true, measureable, or generally-accepted distinction between these two words. Even in the most technical literature the distinction is extremely subtle. For example, in this paper I believe the author is using "sparsity" as a quality or a characteristic of a set (in the boolean sense: either it has it or it doesn't), and "sparseness" as a metric (referring to the "degree of", or "the measure of" sparseness). But even there I think it's a stretch; the words seem to be interchangeable.

Personally, I rarely hear the word "sparsity" used (and only in very technical contexts), "sparseness" being preferred. However, even that distinction doesn't appear to be supported in the literature.

Having said all that, I should point out that in spoken English, neither word is commonly used outside of technical jargon. (Note the relative sparseness of both of the words relative to the word "sparse", for example.) In practice, I would recommend maintaining a sparcity of usage of the words. :)

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