Tap water is man-made (municipally treated) and filtering is a man-made process as well. I guess I'm confused how your water is more inline with nature?
My guess is you probably won't be able to tell much difference since you are adding food to both. Its quite possible the distilled water one will do better but that would depend on the quality of your distillation. A better experiment would be 2 plants outside in the same soil, air, sun. One watered with distilled and one watered with filtered tap.
Many plants also have the capability of converting rock minerals to organic minerals so your tap water plant might do better, but will that prove tap water is better than distilled for human consumption?