Every native speaker has a firm grip on the difference between these two words, but it is hard explain it easily. I won't even try. I'll just point out a few ways they differ:
Example 1:
1) As your father, I recommend that you learn English.
2) Like your father, I recommend that you learn English.
In (1), I am your father. In (2), I am not your father.
Example 2:
1) He walked as a snake crossed his path.
2) He walked like a snake crossed his path.
The first tells you nothing about the way he walked. It only tells you that a snake was crossing his path.
The second tells you that the snake changed the way he walked.
Example 3:
1) She sang as the sun rose.
2) She sang like the sun rose.
In (1), "as" tells you that the singing and the sunrise happened at the same time. In (2), there is no such information. (2) compares her singing to a rising sun.