Groundwater usually begin their life as precipitation: rain, melting snow, etc. These are usually clean and pure (as in distilled) waters and once they percolate through the ground they pick up ions from the rocks and from the soil. For example, groundwater in areas rich in limestone will have high calcium contents. So yes, there are differences in different areas.
Groundwater is mostly safe. However, there are cases where it isn't. Most commonly, groundwater contaminated by man-made sources. Leaks from factories, improper water treatment, etc. Also, some groundwater may be acidic (low pH) or basic (high pH, 1). While this may be desired in natural hot spas, this is not something that would be safe to drink. Groundwater may also be contaminated with arsenic or with radium (2), a radioactive element that occurs naturally in Uranium-bearing rocks (e.g. granite).
References from my local area:
(1) Saines, M., Dickson, P., & Lambert, P. (1980). An occurrence of calcium hydroxide ground water in Jordan. Groundwater, 18(5), 503–503. doi:10.1111/j.1745-6584.1980.tb03427.x
(2) Vengosh, A., Hirschfeld, D., Vinson, D., Dwyer, G., Raanan, H., Rimawi, O., … Ganor, J. (2009). High naturally occurring radioactivity in fossil groundwater from the Middle East. Environmental Science & Technology, 43(6), 1769–1775. doi:10.1021/es802969r