As Kevin said, the higher cohomology groups of constant sheaves on irreducible varieties are zero when working with the Zariski topology. Also, "fibre bundles aren't locally trivial" and "the inverse mapping theorem fails". Instead of writing about the last two myself, let me refer you to the wonderful notes of fellow MOer JS Milne on étale cohomology (page 10, The inadequacy of the Zariski topology)
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