Neat and tidy is an expression that you'll just have to learn and accept. You're correct in saying that either word would be enough to get the thought across, but the two together further clarify the situation by requesting that the form be neat (not crumpled, folded, stapled) and tidy (not stained by your coffee cup or with a cigarette burn on it). As for the official differences in meaning, check your dictionary and thesaurus.
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