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.

When we say that something is in a wrong state, we can only use 間違う.  Examples from [Daijisen](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E9%96%93%E9%81%95%E3%81%86&dtype=0&dname=0na&stype=0) with my English translations:

○ この手紙は住所が間違っている。 (このてがみはじゅうしょがまちがっている。)  This letter has an incorrect address on it.  
× この手紙は住所が間違えている。

○ 間違った考え方 (まちがったかんがえかた)  a wrong way of thinking  
× 間違えた考え方

([○ denotes correct examples and × denotes incorrect examples](http://meta.japanese.stackexchange.com/questions/169/should-we-have-a-standard-notation-for-example-sentences).)