Difference Between Purification and Filtration?

Water Filtration: sand filtrationFiltration is the process of removing physical impurities from the water. The simplest example of a filter would be to pass the water taken from a stream through a fine mesh like a cloth to remove the sand and other sediments. This takes out the actual particles of 'stuff' that is in the water.
Now, filters have become much more sophisticated than this, to the point where filter systems are able to remove actual bacteria from the water. Not because the filter is killing the bacteria, but because the filter has such small pores that the bacteria cannot pass through. Viruses however are too small for the typical kind of filter to remove.

Water Purification: Purification may or may not involve filtration, but will also add the additional element or removing or killing off the viruses. Examples of purifying water without filtration would be boiling, exposing the water to ultra-violet light, or adding iodine or bleach (chlorine) to the water. Notice that in these cases you are killing off the micro-organisms, but you are not removing particles. In fact, boiling water actually concentrates the amount of physical contaminants as water is lost to evaporation but the impurities are left behind.
Regardless of your water source, water purification will be a necessary step. Drinking water that is purified but not filtered may not be pleasant if there is sand grittiness for example, but it won't have the more dangerous elements of harmful bacteria and viruses that could make you very sick.