Reading http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter10.pdf pages 368-370. it states "if we delete the hypothesis that have finite range in the above theorem, then the conclusion is no longer necessarily true".
So my question is
Do you know a (simple?) example of two discrete random variables with finite moments equal for each order but different distributions?
Not quite what you're asking for, but a standard example of continuous random variables with the same moments yet distinct distributions is
$f_1(x) = e^{- (\log{x})^2/ 2} \, / \,(x\sqrt{2\pi})$
$f_2(x) = f_1(x)(1 + \sin(2 \pi \log{x})/2)$
where $ x > 0$.