When you say that: "Esto es igual a esto otro", you are saying that they are similar, or identical, but separated objects. But when you say: "Esta cosa es la misma que la de antes" that means that the thing you are refering too is extactly that, not other, like "this stone is the one we passed before", that stone is "la misma". Besides, "mismo" can also be used to express the same that "igual", it depends on the context and in what do you prefer to use, there is no difference. For example when something have the same size that other thing you can say they have "la misma talla" or that their sizes are "iguales". You just have to be careful with where you place the adjective in the sentence. Hope it was useful for you Michael!