Bottled water stays fresher longer. But yes, "bottling" your own water ahead of time is the next best thing, and a heck of a lot better than going without.
It's also important to fill your bathtub with water, to use as a source of water for uses that don't involve drinking, like rinsing off dishes, and flushing the toilet (dumping a gallon or so of water into the toilet will activate the "flush" sequence even if there's no water pressure in the pipes.
When DSM, Iowa flooded in 93 we lost our water plant for 2 weeks. What saved us is that it continued to rain and we caught water from the downspouts in plastic trash cans and used that water to flush the toilets and for cleaning up. I poured a little bleach in the trash cans every day. My husband actually figured out a camp-type shower for us.