Japan, too, based on my friend's experience there. They use phones for nearly everything, including gaming. But outside of Asia (which is nearly completely separate from the western internet anyway), I don't think many such countries exist. There are many smartphones around, true, but from what I've seen, they're used mostly as, well, phones, incidentally fulfilling other functions. Two major things people do with mobile internet seem to be e-mails and social media, all of which have a dedicated app specifically for that (and nag you to install it if you enter their sites in a mobile browser).
So no, they aren't more important. People read Dilbert, shop, access their bank accounts, read blogs, browse forums, etc. from the comfort (dinky screen is a dinky screen, no matter how you slice it) and safety (as PC security systems are much more extensive than smartphone ones) of their own home. So no. Not more important in the slightest.