His equations work in the limit of slow speeds and low masses; GR keeps giving valid predictions in cases Newtonian mechanics breaks down. Famously, Newton cannot explain the slight observed precession of Mercury’s orbit which is something GR predicts precisely. But the effect is tiny and in almost all cases you don’t need a full relativistic treatment to predict the motions of Solar System bodies sufficiently accurately.

One quite mundane situation where one does need GR is compensating for the clock drift of GPS satellites caused by their height and speed relative to a ground-based observer (time is relative, not absolute as Newton thought!) Because positioning critically relies on very precise clocks, a non-adjusted satellite positioning system would very quickly start giving completely inaccurate results.