Every dollar does count, but being reckless with dollar bills is going to get you in a lot less trouble than being reckless with twenties.
For what it's worth, we have a family of five, my wife stays home, and I believe we currently save somewhere in the ballpark of 35% of our income. I think that counts as being pretty frugal compared to most of the people I know.
Anyways, maybe not that profound, but the memory has stuck. By sweating the cents, you get the dollars. And there’s another saying that is different from yours that I heard my dad say constantly: every penny counts.