The main differences are:
1. Code only vs. code+phase measurements. Measuring the phase of a GPS signal highly increases the accuracy. There are some inaccuracies in measuring the satellite orbits which in turn lead to imprecise receiver position measurements. Then there are (at least on earth) atmospheric efffects and each receiver usually behaves a little different (unstable clocks, multipath effects etc.).
2. single vs. multiple (usually double) frequency receiver. Consumer receivers only measure signals on one frequency while professional receivers measure multiple frequencies. As some of the side effects depend on the signal frequency you can eliminate them by comparing the two signals.

As a result you can get an accuracy that is roughly 100 to 500 times more precise than what consumers experience. We're talking about single centimeters here, for some applications even several millimeters.

Very much.
Another problem is that you have a very small parallax between satellites compared to earth-based measurements.

No way. There are totally different signal types used. Usually lasers and radio frequencies.

It's rust.

They are. Every mars rover is equiped with them. But they are only half as efficient as on earth.