Let me skip to your third paragraph, because this highlights a very important point not commonly appreciated by non-scientists.
In Physics a "theory" is a mathematical model based on various assumptions and valid for a limited range of physical conditions. Newton's laws are a mathematical model that is limited to non-relativistic speeds and low gravitational fields, and within those limits it is exceedingly accurate. There is no sense in which Newton was proved wrong by Einstein. What relativity did is expand the range of physical conditions over which the theory applied. Special relativity extended the range to include high speeds, and general relativity extended it again to include high gravitational fields. Even GR is not applicable everywhere because it fails at singularities like the centre of black holes. We expect that some future theory (string theory?) will extend GR to describe places that are singular in GR.
Anyhow, rant over, and on to your real question. The classic difference is the precession of Mercury. This is probably the biggest effect and it's certainly the most easily observed. Because the orbit of Mercury is an ellipse it has a long axis that points in a particular direction. In Newtonian gravity the direction of this axis doesn't change, but GR predicts it changes by 43 arc-seconds per century. This is a tiny tiny amount. The angular resolution of the unaided human eye is about 1 arc-minute, so you would have to watch Mercury for 140 years before the change in the axis would be perceptible.
(Someone is going to point out this isn't strictly true because the precession of Mercury is about 500 arc-seconds/century, however only 43 arc-seconds of this are due to relativistic corrections. The rest is due to perturbations from other planets accurately predicted by Newton's laws.)