This is a bit of a complex subject.
First: the earth is NOT round it is an ellipsoid.
Second: All Sat GPS coordinates are based on a "Datum" that can date back as far as the early 1920's or even the 1700's or as late as the 1980's There are many datums that can be used.
Also the earth is not "flat" ie. level all around it's surface. It's mountainous, has valleys etc. For GPS to work it must assume that the earth has an average elevation.
This means that for the purposes of "finding" your exact position and translating that to the ancient system of Latitudes and Longitudes means there will be differences from one gps system to the next depending on the "Datum" used.
If my memory serves, Nigel Calders book "How to Read a Nautical Chart" has a thorough explanation of how this all works and it's worth reading for anyone with a gps.
Therefore one GPS will give different sets of coordinates for a given place on the earth than another. Secondly the accuracy of the coordinate system is based upon which part of the earth you are on and how and when the land surveys were done.
My best advice: get and use a paper map. It dispenses with the coordinate system which is unnecessary on land. (for the most part)