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What is the difference in usage of материальная точка and точечная частица? Is the first meant to be merely a simplified model (say, in classical mechanics or relativity) while the second is only for literal point particles? Are they synonymous? Is the second a more recent term?

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"Материальная точка" is a fixed term from Mechanics. Think of material particle/rigid body.

"Точечная частица" is not that important an expression, it is just literally "a particle which is a point: its spacial dimensions are negligible and are considered zero". And here's the trick: right now we use "particle" as a concept in a more recent fields of physics than just mechanics. You can an electron "an infinitesmall particle" but calling it a материальная точка "material particle" will immediately signal that you are viewing an atom strictly as a classical system. That is, from a mechanical point of view.

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