I can be convinced to buy a TV that is twice as expensive if I think it provides the same value. But once you drop down to the 0 mark, it's a different matter entirely (certainly, naive calculations of value break down). Even if you charge for your far superior product, you will have difficulty competing with an inferior free product.
The only real strategy here is to target a niche audience who's needs can't be satisfied with the free product. Many web services take this approach by offering both a free version (for the masses) and a premium one (for the hardcore users), while some solely target the niche (e.g. Smugmug).
But targeting a niche really means that you really aren't competing for the same users as the free product.