Your fish will do fine in either, it is your plants that might need water treatment. It would depend on your tap's PH. If your tap has a PH in the 7-8 range it should be fine to use. Above 8 and you might consider RO or rain water.
An Aquaponic system is best kept at a neutral PH of 7 and will naturally tend to go from a high PH to a lower PH. So if you start at a PH of say 7.8, the natural acidification that occurs as a result from the bacteria will eventually have the PH drop closer to something like 7.
My system has a Ph of around 8 right now, due to my local tap water being highly alkaline. I used tap water but I bought a chlorine/chloramine filter for my hose so I didn't have to aerate out the chlorine before use. Before cycling the water was around 8.4. The change takes place really slowly, but from what I hear, the Ph will eventually end up dropping below the ideal range and you will need to buffer it up.
RO is better IMO. Nothing "extra" for you to worry about.