In certain kinds of abstract thought experiments, it might not match.
If you're doing orbital calculations, it's often wrong.
In the vast majority of real world calculations, it's not approximate. It gives the same answer as relativistic calculations.
Oh, I have an analogy. It's like using 64 bit integers instead of bigints. Any time you input real world numbers, they never go above ten million. Both models give you the correct answer, no error.
No model ever captures the entire nuance of reality. Even a notion of speed is misleading when your objects contain heat. So don't peek inside the model and judge it based on how the pieces interact inside. Judge it based on the results. And Newtonian physics typically give you ideal perfect results.
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Or maybe I should put it a different way:
Newtonian physics are an approximation for the question "In a counterfactual thought experiment where my measurement had infinite precision in trailing zeroes, what would the answer be?"
Newtonian physics are exact for the question "What is the answer based on my measurement?"