Unless you have really bad water straight out of the tap, it's better not to use bottled water. It's usually not necessary, it gets expensive, and a lot of the nutrients you get in tap water are beneficial to your fish and plants in the tank. Unless you have high ammonia, nitrates, or some other pollutant in your tap water, it's not really necessary.
You need a Master Test Kit (like the one from Aquarium Pharmaceuticals, Inc.) and you also need to take some tap water direct from the tap and test it before you add anything to it (no dechlorinating agents ... nothing) for ammonia, nitrites, and nitrates.
If the readings are less than:
then you do not need any kind of bottled water. If the first two are at 0 and you have a Nitrate reading of over 20 but not over 40 you can lower it with the use of the right water conditioner, Amquel+ (it neutralizes ammonia, nitrite, and nitrates) but it should not be started until your tank is cycled as the hampering action toward ammonia will cause a stall in the cycle. (according to the fine folks at Marineland in re: Bio-spira)(but you won't have fish in the tank during the cycle anyway, right?)
If you have Nitrates of over 40 then you will probably need a partial fill of SPRING water, but I agree with Gunnie, not unless it is really needed as you are right there are minerals that are present in tap water that are appropriate for the fish to be getting and it is expensive with any size tank at all.
I used Spring Water in my tanks in the summer when the Nitrate levels around here got above 80 but now I am back to tap water and with real plants (the very best solution for Nitrates) have it down below 20 but in the summer months, I will probably have to go back to Spring Water.
I find that my tap water has a high level of sulphur and phosphates (I had to get special tests from a chemist friend to determine this, the water company wouldn't admit to it) which makes for "green tank" syndrome super fast. Poland Spring Water also tests positive for some phosphates.
As already said, using straight distilled water will lead to pH drop and excessively soft water.
For a while I was using distilled water and buffering the water with pH adjuster and Electroright. This gave me fairly consistent water conditions, but was also an expensive pain in the butt, so I bought a water filter from the LFS for $80 that makes neutral water.
I still have to treat the water with the buffers, but this is overall much cheaper and easier.
RO units are nice, but again, not necessary unless the tap water is really bad. An RO unit will do the same thing as bottled water, plut you have the expense of buying and maintaining the unit. In your case Inuyasha, it would probably be a wise investment since you have a saltwater tank. I don't know what the reasoning is behind it because I don't know much about saltwater tanks, but I do know that a lot of reefers use only RO water for their tanks. My guess is to better control the ammonia, nitrates, and metals in the water going into their tanks.
reverse osmosis requires a pretty big pressure difference. leaving water out is definitely not the same as reverse osmosis. although leaving water out will allow some things, such as chlorine to off gas.