If the polls, poll analysis, and trends are to be believed, Obama needs Ohio much more than Romney does. I don't think Obama can win without Ohio, but I believe that Romney can.
If Obama misses OH, he needs to win every swing state besides NC, VA and FL. So, yeah, Obama needs OH more than Romney needs it. The AP seems to have cooked their “swing state” list for this article to make it seem like Romney needed it more.
Without OH, Obama needs these:
PA
MI
WI
CO
NV
IA
NH