They both are correct but have different meanings. I didn't want this anymore than you did: Meaning both of you did not want it. I didn't want this as much as you did: Meaning both of you wanted it, but the other person wanted it more. Well done! I hope this helps.

They are both ok (except "anymore" should be two words - "any more") but the first sentence means that I wanted this less than or the same as you did and the second sentence means I definitely wanted this less than you did. If you recognise mathematical symbols, it is like the difference between "<=" and "<".