Galileo and Newton correctly thought you couldn't tell if you were moving or standing still if you were in a closed cabin on a ship sailing on a glass smooth sea, but Einstein could explain why you COULD tell the difference if you were sailing in a storm that had huge waves because Einstein realized that the equivalence principle only applies in flat Minkowsky space-time or if the observer is the size of a geometrical point, that is to say had no size at all. Neither Galileo nor Newton ever conceived of curved space-time, or flat space-time, or space-time of any sort. Einstein not only knew that in a gravitational field space-time must be curved he could describe how it was curved with enough details that you could get numbers out of it and check the results experimentally. Einstein also realized the speed of causality must be finite (the speed of light) and thus the rate at which 2 clocks keep time must change depending on their relative velocity.
Of course there are similarities between Galileo and Newton's version of relativity and Einstein's. In 1783 John Michell even predicted black holes, although he called them "dark stars''; he used Galilean relativity to deduce that if a star got massive enough its escape velocity would be greater than the speed of light and thus would go dark; however it had very different properties from a modern Black Hole. If I was far from one of Michell's Newtonian dark stars I could not see it but, unlike a real Black Hole, I could obtain a picture of it and print it in the newspaper, I'd just have to get closer in a powerful spaceship. I could even land on the classical dark star, get a sample of it and then return it to Earth, that sort of thing would be impossible with a real Einsteinian Black Hole.