Does carbonation permanently increase the acidity of soda water?
Both water samples ( assuming pure water ) would theoretically end up slightly acidic with $\mathrm{pH} \approx 5.6$, but the carbonated one after long time. Tap water would have its baseline $\mathrm{pH}$ determined by its $\ce{CO2(aq)/HCO3-(aq)}$ $\mathrm{pH}$ buffer.
This is valid for an ideal case, as side processes - especially for tap water -  will affect the result. As examples come to mind microbial/algae consumption of carbon dioxide, forming compounds affecting $\mathrm{pH}$, or precipitation of carbonates.