In a sense, yes: the strong reaction to Galileo was in part a reaction by a very insecure Church that was trying to contain anything that looked like resistance to Catholic power, because it was facing resistance and reformation on many fronts. Heliocentrism was not inherently a challenge to Church authority except as much as it was mobilized by people who were challenging Church authority. Copernicus' book deliberately bends over backwards to make it clear it was not at all challenging Church authority and doctrine, but was instead resurrecting an ancient theory. Copernicus, it should be remembered, worked for the Church and was inspired to do his work in part because of the call for calendar reform that the Church had already put out.

By Galileo's time things had changed quite a bit. The Church associated heliocentrism with the Reformation. Galileo was certainly swept into this. Copernicus' book was not banned until 1616, and in that same year Galileo was told he was not to hold or defend heliocentric theory, something he agreed to do. He later maintained he was allowed to teach it (in a "teach the controversy" sort of fashion), and he continued to do so. He only really ran into direct conflict, however, when he agreed to write a book (Dialogue on the Two Chief World Systems) on the request of the Pope. He had been asked to write a balanced account; he did not at all, and it was easy to read the book as attacking the Church and the Pope fairly directly. This is what got him into the real trouble. (I am simplifying a lot here, in order to give a brief and direct response.)

So in a sense, yes, the context of the times is why this was an issue in the first place with the Church; a century in either direction and they wouldn't have cared as much. But it is also the case that Galileo himself played his cards very poorly.