The as...as construction is used for comparisons of equality. For example, one could say, "He is as big as I am." It is incorrect to use "like" in these examples because "like" is a preposition and therefore should be followed by an object, not an independent clause. Nevertheless, a LOT of native English speakers use "like" in this way. The correct way to use "like" in my example above would be "He is big like me."