Probably the most famous conflict that satisfies your requirements is the conflict between
"Ptolemy system" and "Copernic system", though few people understand this even today:-)
This conflict became so famous because of the intrusion of a non-scientific authority, that is catholic church. From the point of view of astronomy, there are two aspects:
a) the choice of coordinate system. We are taught in elementary physics courses that this is indeed a matter of choice, of convenience. Some coordinate systems are good for one purpose, others for another purpose. Heliocentric system is by far more convenient when you want to write equations of motion. But geocentric system is more convenient in navigation, where they do not care about "how planets really move" but in "how this looks from the Earth".
b) the more important question whether the orbits are combinations of circles with uniform motion, or Kepler ellipses and the motion obeys Kepler laws. Of course, the discovery of Kepler's laws was a really great discovery (probably more important than the Heliocentric system), because they led to the discovery of the law of gravity, etc., and because they have much simpler mathematical formulation. However when people try to solve the equations of motion for more than 2 bodies, they obtain a solution in a form of
a trigonometric series of functions depending on uniformly changing angles, which is
essentially the same as Ptolemy did.
So you can think of the final result as a representation of the motion as a combination
So in this case we really have a kind of synthesis after a long controversy.
Of course this conflict had also other significance beyond astronomy.
I don't want to discuss philosophical consequences.
The interference of the church into purely scientific matters undermined the church authority, so I would say it lost the cause in the long run.
On the other hand, Galileo is also not so innocent in triggering this conflict. (Copernicus and Kepler did not have any conflict with the church). And the main "proof" that Galileo presented for "moving Earth" was wrong. So it would be much more reasonable on the part of Galileo to hold Copernicus view that the difference between the two systems is mathematical, the choice of the coordinate system, which after all is closer to the truth.
Galileo's main "proof" was based on his totally incorrect theory of the tides which contradicted principles of mechanics he discovered himself in the young age.
EDIT. Some references. They are really abundant (any comprehensive history of astronomy). Some of my favorite in Latin aplphabet:
Simon Gindikin, Tales of mathematicians and physicists. Springer, New York, 2007.
(This is a collection of biographies written by professional mathematician for general public).
Stillman Drake, Galileo at work. A very comprehensive scholarly analysis of Galileo's work and scientific biography, including all controversies. Based on his work and correspondence.
Jean Meeus, Astronomical algorithms, Willmann-Bell, Richmond VA, 1998. How ephemerides
are computed today, with complete formulas for the motion of Sun, Moon and planets, and other related calculations.
John Louis Emil Dreyer, History of planetary systems from Thales to Kepler. Very good book for a beginner. Newer editions are called "History of astronomy from Thales to Kepler"
5.Jean Baptiste Joseph Delambre, Histoire de l'astronomie ancienne, Histoire de l'astronomie du Moyen Age,
Histoire de l'astronomie moderne. These three books are for a serious student of history of astronomy.