No different than using ro/di. If you use tap water you may be putting in things you don't want in your tank (copper, lead, other nutrients and minerals your local water may contain, same if you have a well). Distilled water is pure h20 created pretty much through evaporation. The water evaporates leaving behind all the solids behind that were in it. The evaporated water is collected and becomes distilled pure water. Ro/di water is purified by force through succeedingly smaller mechanical filters and chemical filters to remove the solids but wastes quite a bit to produce the finished ro/di water. Both are just as effective for our purposes the only concern being upfront cost for an ro/di unit, or continual cost for buying distilled water. If your water is like mine and has a tds of ~1100 that would eat up ro/di cartridges, it may be cheaper to just buy distilled or your lfs may sell ro/di for less than a grocery store. Ro/di/distilled is like a blank canvas of water. Whatever you put in it, is what's in it.
I know I said more than you asked but no, distilled should be off no concern at all. I mix my ferts with it. I'm running a 7.5 g saltwater cube with distilled with no issues. So go for it. You might want to add a few ml's of excel to your solutions to prevent mold and stuff but distilled shouldn't give you any issues. I'd also stay away from spring/mineral water for the same reasons you wouldn't use tap.
It depends upon the distillation process used; if the water is distilled using copper piping for the condensation process I would avoid it.
That said I see you are in the Bay Area of CA. Most of the Bay Area water is supplied by rivers and/or reservoirs on rivers flowing out of the Sierra Nevada mountains and is naturally soft. There are a couple areas around the bay that use ground water and that may be harder. Here is a great page showing water sources for various areas around San Francisco; if in doubt call your local water district.
I dose all of my EI ferts dry now but I used to mix them with water and dose the PPM method. I used our local Seattle water (also very soft) and added a few ml of Seachem Excel to the mix to prevent the growth of fungi in the solutions.