They could possible mean the same and also have a different meaning....I suppose its all in the nuance.... imho I equate the "much" with quantity and "often" with frequency.....!
Hope I'm not being rude keitra, perhaps you're not a native English speaker but, whichever you choose (and I'd go for "often"), you should probably use a definite article in front of "Internet" i.e "I use the Internet very often".
Oh that reminds me, does any one know, in Arabic the word for more as in "more tea" meaning stronger tea is different to more in 'more tea' meaning another cup.
As others have said, "very often" would sound more natural here. But in the negative, it would sound fine to say "I don't use it very much." I think BlueDolphin is correct in the distinction between quantity (I like it very much) and frequency (I use it very often). "Use" seems to refer to individual instances of use (9 times), whereas "like" can't be measured in units. But you can use "much" to ask, "Do you use the Internet very much?" I guess that's because you're not referring to individual instances, but rather an unknown quantity....?