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So I know that Einstein and general relativity had huge impacts on the way we view the world, but how crucial were these scientific advancements to the success of our space programs? Would Newtonian physics sufficed, or would using those formulas and methodologies when travelling off-earth and to the moon resulted in catastrophic failures?

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This blog article looks relevant, though the pictures are broken at the moment. –  John Rennie Mar 13 at 16:22
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This question appears to be off-topic because it is about space travel. Space.SE might be a better place for this question. –  Kyle Kanos Mar 13 at 17:25
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Surely there is a valid physics answer here: simple calculations of paths in inverse square field and compare the path with a geodesic in the Schwarzschild metric? So to and from the moon, no, there is no practical difference: if you want to know the position of a space station orbiting at Mercury's distance from the Sun and you're using this space station as a positioning system beacon, then yes absolutely you'll need to use the GR calculations. It all depends on your application, but your question is a good one, so I don't agree with the close votes. –  WetSavannaAnimal aka Rod Vance Mar 14 at 21:47
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Well, relativistic effects are accounted for in all space travel. These effects were apparently minor enough during the Apollo missions to rely on standard Newtonian equations and minor course corrections enroute, but are pronounced enough on longer missions within the solar system to account for several kilometer errors.

General relativity is routinely accounted for in spacecraft navigation.... the NASA navigation software developed at JPL....incorporates the Ted Moyer formulations for navigation, which includes mathematical expressions that describe the effects of general relativity.

References
RELATIVISTIC TIME CORRECTIONS FOR APOLLO 12 AND APOLLO 13
Accounting for General Relativity at Mercury

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That's actually the opposite of what I've heard. Do you have sources? –  David Z Mar 13 at 16:18
    
I'm looking for some. –  Robert Harvey Mar 13 at 16:19
    
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Not to be a pain, but that file seems to be analyzing how a clock carried on the spacecraft would differ from one kept on Earth. It doesn't say anything about the trajectory of the spacecraft, as far as I can tell, which it would need to in order to be relevant to this question. –  David Z Mar 13 at 16:38
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It would have been a relatively small effect for moon missions, given the ability to navigate and apply small course corrections. –  Robert Harvey Mar 13 at 17:21
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