To begin, one must understand the difference between denotative and connotative meaning. First, a denotative meaning refers to a word's dictionary definition. For example, the dictionary meaning of sleep, according to Merriam-Webster, is as follows:
the natural periodic suspension of consciousness during which the powers of the body are restored.
Connotative meaning, on the other hand, is very different. The connotative meaning of a word refers to one's emotional definition. It is, then, the meaning an individual applies to a word, outside of the denotative meaning.
In regards to the connotative meaning of sleep in William Shakespeare's Macbeth, each instance of its use has different meaning for both the speaker and the reader.
In the first quote, "The sleeping and the dead are but as pictures" (II, ii, 68-69), Lady Macbeth is trying to convince Macbeth that there is no difference between one who is sleeping and one who is dead. Macbeth has just stated that he is far too afraid to reenter the chamber after murdering the king's men. Therefore, the connotative meaning of the word sleep, here, is meant to parallel the meaning of the word sleep; essentially, this shows that there is no difference between death and sleep.
In the second quote, "Shake off this downy sleep, death's counterfeit, And look on death itself!" (II, iii, 81-82), Macduff is telling the men around him to wake up. Here, the use of the word sleep is placed in direct opposition to Lady Macbeth's use in the previous scene. Macduff does not consider sleep to be similar to death. Instead, he believes that sleep is very different from death, given one cannot wake from death (as they can from sleep).
In the end, the connotative meaning of sleep differs greatly between Lady Macbeth and Macduff. While each recognize sleep exists, they each define it in an emotional way. Lady Macbeth is far more unemotional about death. Macduff,on the other hand, reacts very emotionally to the word. He, unlike Lady Macbeth, understands and recognizes the reality of death.