Much blame has been directed at [Pope] Urban for the Inquisition’s condemnation of Galileo and the prohibition of his ‘Dialogue on the Great World Systems’. In fact he cannot reasonably be reproached for his attitude toward the affair, which was always one of studious moderation. When, in 1615, Galileo first got into trouble for his advocacy of the Copernican system, we find Matteo Barberini, at that time a Cardinal, declaring his admiration for the great mathematician but warning him not to fall foul of the theologians. It was partly owing to his intervention with Paul V that the Holy Office’s decree of 1616 denouncing the heliocentric theory was followed, so far as Galileo was concerned, by a new admonition from Cardinal Bellarmine. Otherwise he was not punished.
There was never any danger of his being sent to the stake like that earlier victim of Bellarmine, Giordano Bruno. But Bruno, the apostate Dominican whose free-thinking philosophy cut at the roots of Catholic doctrine, was in a very different category from the astronomer who, in Barberini’s view, was doing no worse than propounding a brilliant hypothesis.
Urban’s friendship with Galileo survived the admonition. In 1633, when Galileo was hauled before the Inquisition for the second time, the Pope took the utmost care to ensure that he was leniently treated, that he was not put in prison and that his trial and abjuration were so managed as to allow him to stay unharassed and in comfortable retirement for the remaining nine years of his life.
Sir Nicholas Cheetham, Keepers of the Keys: A History of the Popes from St. Peter to John Paul II (1982)
There are some additional points to make about Giordano Bruno, with whom Tumblr has a bizarre fascination.
He was ordained a Catholic priest when he was a young man, but his heresy became evident a few years later. After fleeing from his Dominican Order and his charges of heresy, he was (in no particular order):
excommunicated by the Catholic Church
and by the Calvinists
and by the Lutherans
and denied from lecturing at Oxford
When he was denounced to the Venetian Inquisition and later extradited to the Roman Inquisition, his charges were not about his science and heliocentrism (which was terrible), but about his faith (which was openly heretical). The Catholic Church didn’t even have a position on heliocentrism yet. That would come about 30 years later, with Galileo - who got in trouble because he argued heliocentrism was truth, rather than hypothesis.
Bruno was kept in prison before his trial for roughly 6 years - it is hypothesized that Church and state officials spent that time encouraging him to recant. After a trial in 1599, he was condemned to death for heresy against the Catholic Church, since he was a Catholic priest who had actively and knowingly preached anti-Catholic teaching.
It is critical to note that inquisition tribunals were an interesting mix of Church and state power. The Church did not retain the right to put people to death. Parties condemned to death were turned over to the appropriate secular power, who could decide to carry out the judgment. Before the French Revolution, religion and politics were knit closely together - heretics of the national religion were typically seen as enemies of the state as well.
It has been remarked that he was a poor theologian and an even poorer scientist - it is nearly impossible to make sense his his pseudo-scientific works. Bruno should be found offensive not only by Catholics, but by Calvinists, Lutherans, scholars, and scientists.