There is an important distinction: lo mismo means that the two things are actually the same object (or idea, or person, or whatever), while igual means that they have many characteristics in common. For example, say you are walking down the street and see a random person. The next day, you happen to see that person again. You'd probably think "Debe de ser la misma persona", that is, it probably was the same person you see both days. However, if you somehow found out it was actually a pair of twins you saw, you'd think something like "Con razón; eran iguales", meaning "No wonder; they looked the same".
In short, lo mismo is about identity, igual is about appearance or characteristics. Let's say your grandma says "Estás igual a tu padre" (You look just like your father). It makes sense, because igual doesn't mean you are the same person, it just means you look alike.
As a further example, say a couple is getting divorced. The man asks why, and his wife says "No eres la misma persona que cuando te conocí" (You are not the same person as when I met you). Physically, the man may look just like the time they met, but he may have changed in other ways, and therefore is not considered the same person.