Both "as" and "like" are used to make comparisons, but they are used differently.
These are good sentences meaning that Mary is faster than me:
"I can't run as fast as Mary runs"
"I can't run as fast as Mary"
"I can't run fast like Mary runs"
Bad sentence: "I can't run as fast like Mary".
Bad sentence: "I can't run fast as Mary runs".
Observations:
"As" sets up a comparison between two things. These two things need to be clearly labeled, and this is best done by using the form "as ... as ..."
"Like" sets up no such comparison. For this reason, the sentence
"I can't run fast like Mary"
is impossible to interpret. It could be have two completely opposite meanings. It could mean
(1) Mary runs faster than me, or
(2) I don't run fast, and neither does Mary.
You can use "like" to create the comparison, but you have to change the word order:
"Like Mary, I can't run fast"
so now it clearly means that both people are slow runners. Or, to use "like" to say that Mary is faster you could say
"Unlike Mary, I can't run fast".