**** Etymological explanation - 無理： 理が無い（ので達成できない） you can't acieve it because it is unreasonable - 駄目： it makes no difference (やっても無意味、無価値） This word originates from GO (囲碁) game, 駄目 in 囲碁　means "useless/worthless slot", which turns to mean simply "useless/worthless". ＊＊＊＊＊ However, nowadays 駄目 is often used to express prohibition, "Don't do that" 「だめ、来ないで」 「薬物濫用。ダメ！ゼッタイ　http://www.dapc.or.jp/」 you usually use むだ to express "worthless/useless even if you try". ＊＊＊＊ example このプールは深いから、子供が泳ぐのは無理だ。: A description with a reason. Children can't swim here because they are small but the pool is deep. このプールは深いから、子供は泳いじゃダメ : A prohibition Children is not allowed to swim here because it is dangerous.

無理な ≒ impossible/unreasonable 駄目な ≒ bad/useless