The deflated balloon has only the weight of the balloon skin, but the inflated one has the weight of the balloon skin and the air inside. This makes it heavier because air has a weight, even if it is very small it has a weight. If you filled it with helium, you might then think it was lighter but still a balloon filled with helium is still heavier than the deflated one because helium is less dense than air (since it rises) but still has a weight.
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