Difference between tf-idf and tf with Random Forests
Decision trees (and hence Random Forests) are insensitive to monotone transformations of input features. 

Since multiplying by the same factor is a monotone transformation, I'd assume that for Random Forests there indeed is no difference. 

However, you eventually may consider using other classifiers that do not have this property, so it may still make sense to use the entire TF * IDF.