The relativistic corrections are tiny and in the early 20th century, only general relativity was found to matter, and it only matters for the precession of Mercury's perihelion.
But your words indicate that you are neglecting all deviations from circular 2D orbits predicted by Newton's theory.
First, the problem isn't 2D. It's 3D and the planes of the orbits don't coincide. Relatively to the Sun-Earth orbital plane (the ecliptic), the orbital inclinations (angles by which the planes are tilted relatively to each other)
go from 0.77 degrees for Uranus to 7.01 degrees for Mercury. The latter is a pretty big angle. When the road is inclined by this angle, one may easily feel that it's not flat in the car. The nonzero inclinations are the reason why we don't get the lunar and solar eclipses every month (each of them), not to mention all the other "syzygies" (three celestial bodies are exactly in line).
Then you are neglecting the fact that even for the Newtonian 2D problem, the orbits are not circular but elliptic. Eccentrities – the relative numbers indicating how squeezed the ellipse is, or how far from a circle
are as high as 0.09 for Mars, 0.21 for Mercury, and 0.25 for dwarf planet Pluto. Even Mars' eccentricity 0.09 is so big that you would have trouble to find the planet in the telescope if you assumed the circular shape. And even Earth's 0.017 is enough to substantially contribute to the asymmetry of the seasons. Antarctica has seen cooler record low temperature than the Arctic partly because during the Southern winter, in July, we are further away from the Sun. (Another reason is the Arctic Ocean: the water reduces the overall magnitude of temperature fluctuations.)
Well before people believed that the Sun was at the center of the Solar System, they were highly aware of the non-circular shape of the orbits. They described them by the so-called epicycles (circles on top of circles). It's both simpler and more accurate to describe them as ellipses that obey the three Kepler's laws
Kepler's laws say that the orbits are elliptic; the Sun is in one of the focal points; the bigger semi-axis and the orbital period scale according to a power law; and the speed of the planet is variable but such that the "speed in terms of the area per unit time" is constant for each planet.
If you described the planetary orbits as inclined i.e. 3D elliptic orbits obeying Kepler's laws, you would be very close to the reality. The biggest omission is the interaction in between the planets themselves, especially Jupiter's action on other planets. The elliptic orbits assume that the Sun is the only object that acts on a given planet. Jupiter adds a significant correction which modifies the shape from exact and periodic ellipses to something slightly aperiodic and even more slightly different from ellipse even locally.
One must also realize that the center-of-mass of the Solar System isn't exactly at the center of the Sun. It's at the "barycenter" which is slightly off the center of the Sun but because the planets are light enough, it's still within the Sun's volume.