"Compared with" vs "Compared to"—which is used when?

From Strunk and White:

To compare to is to point out or imply resemblances between objects regarded as essentially of a different order;

To compare with is mainly to point out differences between objects regarded as essentially of the same order.

Thus, life has been compared to a pilgrimage, to a drama, to a battle; Congress may be compared with the British Parliament. Paris has been compared to ancient Athens; it may be compared with modern London.

Use "compared with" when you are looking for differences.

E.g. CEO’s now earn 419 times the pay of blue-collar workers, compared with 42 times their pay in the 1980's.

Use "compared to" when highlighting (or comparing) the similarities of one thing to another.

E.g. The human heart can be compared to a pump.

Bryan Garner, Garner's Modern American Usage, fourth edition (2016) provides what I take to be the current (and traditional) formal prescriptivist view among U.S. usage authorities of when to use compered with and when to use compared to:

compare with; compare to. The usual phrase has for centuries been compare with, which means "to place side by side, noting differences and similarities between" {let us compare his goals with his actual accomplishments}. Compare to = to observe or point only to likenesses between {he compared her eyes to limpid pools}.

But one of the most familiar "comparisons to" in English literature, Shakespeare's Sonnet XVIII, appears to flout that supposed rule in the first two lines:

Shall I compare thee to a summer's day?

Thou art more lovely and more temperate:

If we take Garner's rule at face value, Shakespeare, since he wasn't merely saying "You are like a summer's day," should have said

Shall I compare thee with a summer's day?

He puts the day and the beloved side by side and promptly concludes that the beloved is more lovely and more temperate. Those, in a nutshell, are Garner's conditions precedent for using compare with. And Shakespeare can't argue that "The meter made me do it," because with is just as unstressed as to. Either Shakespeare bricked his choice of preposition or this centuries-old rule is less honored in the real world than Garner and others would have us believe.

There is a third possibility, though: In line 1 of the sonnet, Shakespeare may be using compare to in its approved sense of "liken"—in effect saying, in the first two lines, "Shall I liken you to a summer's day? Nah, you're better than a summer's day." But this interpretation takes us only so far: by saying in line 2 that his beloved is more lovely and more temperate than (that is, different from) a summer's day, Shakespeare is unmistakably comparing the beloved with the summer's day, according to Garner's guidelines for distinguishing between compared with and compared to.

An Ngram chart of "compared to" (blue line) versus "compared with" (red line) versus "in comparison to" (green line) versus "in comparison with" (yellow line) shows considerable change in the phrases' relative frequency of use in published writing since 1920:

The lines in the graph show compared with breaking away from compared to in frequency of use around 1760 and then steadily increasing its advantage until about 1920; then compared to begins a very brisk ascent in frequency that catches up to compared with (which has been in something of a decline since about 1960) around 1980, and continues to put distance between it and its rival over the next 25 years. Meanwhile, in comparison with opens a small but consistent lead over in comparison to by 1800 and for the next 140 years increases the advantage. But around 1940 the two lines begin converging, and at about the turn of the millennium they meet.

The line graphs in the Ngram chart suggest either that people now use compared to in the sense of "liken" far more often than they did a century ago or that the rule specifying when to use compared with and when to use compared to is fading from actual usage in published content. I think that the second explanation for the change in the Ngram data is far more likely than the first to be true.

It is certainly possible that the preference for compared to in instances in which the meaning is "likened" remains fairly strong, although there is no easy way to confirm this possibility. But the preference for compared with in instances in which the meaning is "examined side by side" seems to be in decline. The confluence in frequency of "in comparison with" and "in comparison to" (also shown in the Ngram chart above) offers some circumstantial weight to this hypothesis, since "in comparison to" seems never to have had a "liken" sense, as "compared to" sometimes does.

I expect the trend away from observing and enforcing the traditional compared with/compared to distinction in connection with the meaning "examined side by side" to become even more general as the involvement of copy editors in the publishing process continues to diminish.

We compare with when we are going to look for differences...My English is better compared with it was in 2013...that is,I can see a difference in my English....

We compare to when we see similarities... Rio De Janeiro City is compared to California... This sentence means that we can find some similarities between Rio De Janeiro and California.