Question: I had a friend who said there is a slight difference in the two phrases. But I don't see any difference between the two expressions. Could anyone explain it? 

Thanks,
Rachel

Answer: 
Let's say you went to a nice restaurant for a nice meal. 

Case 1: you had the meal, course by course. In this case 最后吃的菜=(吃)最后的菜=dessert

Case 2：you left in the meal for some reason. In this case 最后吃的菜=the dish you didn't finished. (吃)最后的菜=dessert.
You need to put these statements in their context to be certain of their exact meaning. Without further context, I would say:

最后吃的菜 = the (very) last dish(es) I ate (of a meal)

吃最后的菜 = the last dish(es) (of a meal which) I am eating/I haven't eaten yet