Spring water is term applied (too liberally) to a body of water, where the main supply is a natural infiltration of water from various other sources. Some “spring” waters claim to be more pure because they come from the springs which are formed from snow melt. However, as hard as it may be to imagine, snow and rain are both dirty. Consider this; it is standard to accept that if you wish to make a poor quality source of water better quality, you should boil it. The length of boil should be roughly 10 to 15 minutes, depending on the quality of the source. However, rain (and snow) is the product of solar distillation. The sun will very rarely heat water to 100°C, and in the cases of mountains, where water will boil (and thus evaporate to the atmosphere to form rain and snow) at lower temperatures, the removal of bacteria, viruses, protozoa, and nasty chemicals will be quite low.
Add this to the fact that the rain or snow must fall through the atmosphere, where there is a lot of suspended material (particularly particle sizes of less than 10µm, known as PM10), that the rain and snow can gather as it travels to the “spring”. Once in the spring, there are undoubtedly other sources of inflow, such as runoff from the banks, where there will most likely be some form of life, life that needs water to live, so it goes to the spring to drink, since its nice to eat as well, the life will most likely have some dinner too.
This life could be anything, ranging from huge mammals, to tiny insects. All produce excrement, of which, mammalian is the most likely to carry the protozoan threats; Cryptosporidium sp, Giardia sp, Clostridium perfringens and various other very nasty water borne diseases (typhoid, cholera, polio). However, bottled water manufacturing is (at least in the state of Queensland, Australia) governed by the use of a HACCP system. This means that any bottled water has to have been treated to safely reduce any identified risks in the source. For example, if some monitoring found that there was some Giardia sp in the source, then the risk would need to be mitigated by filtration. If some E-coli or F-RNA phages were found in the source, then the risks would need to be mitigated by; chlorination or chloramination, or ozonation, or micro-filtration.
The point is, nearly all natural “springs”, even pristine catchments, will contain life, which must produce excrement, which could wash in to the spring, which brings all of the above risks to a head. This means, that in order to sell a natural spring water in a bottle, the manufacturer must treat the water, and in some instances, treat more risks than in ordinary tap water. An ordinary drinking water plant will just have a set of filters and chlorination. However, bottled water will almost certainly never use chlorine or chloramination to treat bacterial risks, they will use other methods that are less easily detected by human senses (such as ozone or micro-filtration). Micro-filtration will not remove ionic compounds out of the source water.
In conclusion, the difference between tap water and spring water? Tap water will withstand a longer amount of time out of the fridge than bottled water.
There have been cases documented where bottled water has reportedly had a film of growth – this is harmless and is simply due to the fact that there is no free chlorine to stop the growth occurring. This won't happen with tap water though. With chlorine, I guess you get other risks too, such as halo-aceto nitriles, halo-acetic acids and tri-halo methanes. However the biggest risk from these compounds is from inhalation of the vapor, and this will most likely occur in showers, baths, anywhere there is hot water really.