違う and 間違う: a difference and a mistaken choice
Distinguish a neutral difference from an error against a standard, and use conjugation to recognize 違う as a verb rather than a na-adjective.
Different does not necessarily mean wrong
私と姉は好みが違います。 says that the speaker and their older sister have different tastes. 違う, read ちがう, compares the two without declaring either one wrong.
昨日とは違う道を通りました。 means that someone took a different route from yesterday. 違う directly modifies 道; do not insert な to make 違うな道.
An incorrect answer or choice
その答えは違います。 can mean that an answer is wrong when checking against a correct answer. Whether it marks difference or error depends on the standard being used.
道を間違って、駅と反対の方向へ歩きました。 says that someone took the wrong route and walked away from the station. 間違う, read まちがう, describes a mistaken choice or error.
Negative forms and correction
違う has the negative form 違わない and the past form 違った. 大きくは違いません。 means that the difference is not large; it does not necessarily deny every difference.
番号が間違っています。 points out that the number is wrong. Following with 正しい番号を確認してください focuses on checking the correct number rather than judging the person.
Whose taste is wrong in 私と姉は好みが違います?
Show the answer and why
Neither person's taste is said to be wrong. It compares their tastes as different without supplying a standard of correctness.