If your going PC for the grandmother, go with the AMD based system. An Atom (even a dual core one) is going to be crushed by a proper desktop CPU, even if it is a low end Athlon. A quick search showed the Atom 330 hitting around 1950 in PCMark 05 CPU testing, while the Athlon 3250e hit 2919. For comparison, a 2.26ghz Core 2 Duo hits a bit above 4000.

Keep in mind the low spec now is going to mean a really low end spec a year, two or three from now. Things like Hulu and such are unplayable on a low end machine from just three years ago. I have no idea if your grandmother has need of such internet services, but going low end now may mean an upgrade is needed sooner for them down the road.