In Sonnet 116, the speaker uses metaphors to compare Love to

an ever-fixed mark
That looks on tempests and is never shaken;
It is the star to every wand'ring bark,
Whose worth's unknown, although his height be taken. (lines 5–8)

In other words, Love is constant and unwavering, despite how lost one might feel or how far one may travel. Love is consistent, its movements predictable like a star's. Neither can it be destroyed nor changed by the most violent of storms. This idea presents another metaphor, that of troubles or challenges in a relationship to physical, weather-related storms, or "tempests."

In Sonnet 130, the speaker makes fun of the typical comparisons used by lovers to describe their ladies. It is not unusual for a lover to flatter the woman in which he takes an interest, and the speaker takes this practice to task because it does a disservice to real love. So, the poem begins with a kind of reverse simile, where the speaker emphasizes the humanity of his lover by stating that her "eyes are nothing like the sun." Her lips are not like coral, her skin is not like snow, though he uses a rather unflattering metaphor to suggest that "black wires grow on her head" (4). It is not that he wishes to insult his lover; in fact, it's just the opposite. He claims that genuine, real love doesn't need to use false comparisons to prove how rare and special it is.

In Sonnet 18, the speaker compares his lover to a "summer's day" via a series of comparisons where, as it turns out, her beauty is actually superior. He employs a metaphor to claim that her "eternal summer shall not fade" (9). Her beauty and youthfulness will never die because he has immortalized it in words, and so long as people exist, "So long lives this [poem], and this gives life to thee" (14).

In Sonnet 29, the speaker feels pretty dejected, envying other men their accomplishments and happiness. However, he claims, when he remembers the "sweet love" of his mistress, he feels a change within himself "Like to the lark at break of day arising / From sullen earth" (11–12). In this simile, he claims that his spirits rise like the lark who flies from the earth when day begins. When he recalls that he is loved by her, he no longer feels sad and wouldn't even change places with a king.

At the outset, let me restate the difference between these two terms: metaphor is comparing two traits or items with some similar qualities (but not identical); simile is a kind of metaphor, one that uses “like” or “as” to make the comparison.  The term “figure of speech” covers many devices available to stretch the language past literal meaning.  Take, as just one example, personification (“the sun smiled on me"), in which a nonhuman object takes on human characterizations.  

 Sonnet 116 compares unmoving, constant things—an ever-fixed mark, a star, etc.--to the “marriage of true minds,” meaning that it does not change when circumstances change..

 Sonnet 130 uses simile (directly using “like”) to enumerate a list of “false comparisons,” exaggerations usually found in false lover’s descriptions, to stress his sincerety in contrast to the superficiality of her “false compare.  Shakespeare cleverly criticizes metaphor and its users by exposing its exaggerations to contrast with his understated “rare” love.

 In Sonnet 18 he uses the opposite strategy, finding imperfections in the comparisons that are ready to hand by mediocre lovers.  He objects to standard metaphors because they are all imperfect, and his lover is not.

In Sonnet 29, not till line 10 “and then my state/Like to the lark arising…” does a direct comparison (simile) appear.  The majority of the poem is a description of his dissatisfied state, ameliorated by remembering his “sweet love.”