In comparisons, "like" only express similarity, whereas "as...as..." makes a more precise comparison that can even contrast degree or quantity. For example, consider
A) The stones were hot like fire. (similar to fire)
B) The stones were as hot as fire. (as much as fire)
#A says that the heat of the stones was similar to a fire, more or less. #B says that the stones were at least as hot as fire.
The imprecision of "like" often makes it the better choice in poetry:
"her voice was soft like summer rain"
in which "as" would ruin the image with its precision.
There are situations where one word makes sense, but not the other. For example,
"As far as I know, today is Wednesday"
makes a comparison between what I know and the limits of my knowledge. It concisely states that what I know to be certain is limited by my knowledge, sort of a strange thing to say, but everyone intuitively grasps what it means. I cannot imagine a way of saying this using "like". It would require a whole paragraph.