I would say that the Asus eeeBox's performance will be better than the good ol' Athon 1.2GHz. Without a doubt. The Athlon 1.2GHz itself is considered to be anemic nowadays.
It's not something I would buy for myself, but it has a nice sleek design that your wife will probably appreciate. It takes up very little desk space and everything is easily accessible. The only real problem I have with it are the open ports / slots which can allow dust to collect up over the years. Solvable by simply wiping it every now and then, but some people tend to end up being lazy and eventually stop doing so.... (like me).
You should note that the keyboard is not full size, there is no numeric pad. But if your wife is only doing word processing and web surfing, then it's not be big issue.
Just in case, it is not meant for gaming. I remember reading about someone trying to install the Sims 2 and he/she got some type of video error message when trying to run the game.