http://briankoberlein.com/2014/08/05/small-changes/
Brian Koberlein: "One aspect of general relativity that always amazes me is the level of precision needed to distinguish it from Newtonian gravity. Take, for example, the advance of Mercury's perihelion. When you count in the gravitational tugs from the sun and all the planets, Newton predicts Mercury's perihelion will advance about 531.65 arcseconds per century. When we measure the orbit of Mercury, we find its perihelion actually advances 574.10 arcseconds per century. This means Newton's prediction is off by about 42.45 arcseconds per century. I say "about" because there is an uncertainty in our observations of about 0.65. General relativity predicts an "extra" perihelion advance of 42.98, which agrees exactly with experimental observation. The difference between Newton's model and Einstein's amounts to 28 millionths of a degree each orbital revolution. Put another way, Mercury makes one orbit every 87.969 days, but it reaches its perihelion about a half second later than Newton says it should. The difference between Newton and Einstein is less than a human heartbeat in time."

The explanation is simple, Brian Koberlein. Newton's prediction is "off" because some mass is either unaccounted for or assumed to be in the wrong place:

http://irfu.cea.fr/Phocea/file.php?f...TE-052-456.pdf
Jean-Marc Bonnet-Bidaud: "L'épilogue du dernier test de la relativité, celui de l'orbite de Mercure, est encore plus passionnant. Ce fut en réalité un test a posteriori de la théorie, puisque la prédiction a fait suite à l'observation et ne l'a pas précédée. L'accord est stupéfiant. Le décalage observé dans la position de Mercure est de 43,11" par siècle, tandis que la prédiction de la relativité est de 42,98" par siècle ! Cette révision de l'horloge cosmique est toujours considérée comme le grand succès d'Einstein, mais elle est encore sous l'épée de Damoclès. En effet, des scientifiques soupçonnent que le Soleil pourrait ne pas être rigoureusement sphérique et un "aplatissement" réel introduirait une correction supplémentaire. La précision actuelle deviendrait alors le talon d'Achille compromettant le bel accord de la théorie."

On the other hand, Einstein's prediction "agrees exactly with experimental observation" because Einstein changed and fudged equations until "excellent agreement" was reached:

http://www.weylmann.com/besso.pdf
Michel Janssen: "But - as we know from a letter to his friend Conrad Habicht of December 24, 1907 - one of the goals that Einstein set himself early on, was to use his new theory of gravity, whatever it might turn out to be, to explain the discrepancy between the observed motion of the perihelion of the planet Mercury and the motion predicted on the basis of Newtonian gravitational theory. (...) The Einstein-Grossmann theory - also known as the "Entwurf" ("outline") theory after the title of Einstein and Grossmann's paper - is, in fact, already very close to the version of general relativity published in November 1915 and constitutes an enormous advance over Einstein's first attempt at a generalized theory of relativity and theory of gravitation published in 1912. The crucial breakthrough had been that Einstein had recognized that the gravitational field - or, as we would now say, the inertio-gravitational field - should not be described by a variable speed of light as he had attempted in 1912, but by the so-called metric tensor field. The metric tensor is a mathematical object of 16 components, 10 of which independent, that characterizes the geometry of space and time. In this way, gravity is no longer a force in space and time, but part of the fabric of space and time itself: gravity is part of the inertio-gravitational field.. Einstein had turned to Grossmann for help with the difficult and unfamiliar mathematics needed to formulate a theory along these lines. (...) Einstein did not give up the Einstein-Grossmann theory once he had established that it could not fully explain the Mercury anomaly. He continued to work on the theory and never even mentioned the disappointing result of his work with Besso in print. So Einstein did not do what the influential philosopher Sir Karl Popper claimed all good scientists do: once they have found an empirical refutation of their theory, they abandon that theory and go back to the drawing board. (...) On November 4, 1915, he presented a paper to the Berlin Academy officially retracting the Einstein-Grossmann équations and replacing them with new ones. On November 11, a short addendum to this paper followed, once again changing his field equations. A week later, on November 18, Einstein presented the paper containing his celebrated explanation of the perihelion motion of Mercury on the basis of this new theory. Another week later he changed the field equations once more. These are the equations still used today. This last change did not affect the result for the perihelion of Mercury. Besso is not acknowledged in Einstein's paper on the perihelion problem. Apparently, Besso's help with this technical problem had not been as valuable to Einstein as his role as sounding board that had earned Besso the famous acknowledgment in the special relativity paper of 1905. Still, an acknowledgment would have been appropriate. After all, what Einstein had done that week in November, was simply to redo the calculation he had done with Besso in June 1913, using his new field equations instead of the Einstein-Grossmann equations. It is not hard to imagine Einstein's excitement when he inserted the numbers for Mercury into the new expression he found and the result was 43", in excellent agreement with observation."

Pentcho Valev