like => followed by a noun phrase: - This coffee tastes like mud. - He runs like a racehorse. as => followed by a verb phrase: - This coffee is as muddy as it was yesterday. - He runs as though he were a racehorse. Or, in Colton's 2nd example: - You are as dumb as a box of rocks [is]. The verb "is" is usually omitted as understood.

Like is, when you compare it to as, usually used to compare two things and say that they are alike. That car looks like a bug! As is used for comparing two things in a different way. "You are as dumb as a box of rocks!" Its is saying that you (not actually you this is just the sentence I thought of haha) are equally dumb as a box of rocks. Does that help?