Almost same but these two words have different connotations. For example 이건 내 '잘못'이야. =this is my fault. 이건 내,'실수'야= this is my mistake.

check this one as well - <a href="http://koreanselfstudyisntlame.blogspot.com/2010/07/vs_31.html">실수_vs_잘못</a>