When we say that someone makes a mistake about something, we can use both 間違う and 間違える and there is no difference in meaning or nuance. For example, 計算を間違えた and 計算を間違った mean the same thing.
[Added remark: As repecmps pointed out in a comment, some people consider this usage of 間違う as incorrect. Someone states that this usage of 間違う was originally incorrect, but I do not know if this statement is true or not.]
When we say that something is in a wrong state, we can only use 間違う. Examples from Daijisen with my English translations:
○ この[手紙]{てがみ}は[住所]{じゅうしょ}が[間違]{まちが}っている。 This letter has an incorrect address on it.
× この手紙は住所が間違えている。
○ [間違]{まちが}った[考]{かんが}え[方]{かた} a wrong way of thinking
× 間違えた考え方
(○ denotes correct examples and × denotes incorrect examples.)
