Merriam-Webster's view of how sympathy and empathy differ has evolved over time. In the past eighty years, MW has attempted on three occasions (that I'm aware of) to distinguish between the two terms, and each time it has thoroughly revamped its explanation.
'Sympathy' and 'empathy' in Webster's Dictionary of Synonyms (1942)
MW's Webster's Dictionary of Synonyms (1942) addresses empathy and sympathy in a bundle of synonyms headed by pity:
Pity, compassion, commiseration, ruth, condolence, sympathy, empathy, bowels agree in meaning a feeling for for the suffering, distress, or unhappiness of another. ... Sympathy (etymologically, suffering with) is often used in place of pity or compassion (as, his plight aroused her sympathy) or in place of condolence (as, to offer one's sympathy to a bereaved friend), but in its precise meaning, it implies a power to enter into another's emotions or experiences, whether of a sorrowful or joyful nature, as by sharing them, by truly understanding them, or by being equally affected by them; as, "a boy goes for sympathy and companionship to his mother and sisters, not often to his father (A. C. Benson); "the rebel, as a human type entitled to respect and often to sympathy" (R. E. N. Dodge); "Amid the various feelings she was aware of arousing, she let me see that sympathy, in the sense of a moved understanding, had always been lacking" (E. Wharton); "Ah, then that was it! He was a lonely old man, who didn't want to live in constant reminder of happy times past. ... Tony ... felt a quick sympathy with him" (Arch. Marshall). Sympathy is also applicable to to anything that engages one's interest, sometimes because one is in agreement with its aims, accomplishments, principles, or tenets, and is attached to it (as, "the stepfather was a moderate Pompeian in sympathies"—Buchan), but more often because one has the imaginative capacity to enter into it and understand it in its true nature (as, "a creative writer can do his best only with what lies within the range and character of his deepest sympathies"—Cather). Empathy applies to the imaginative power which enables a person, especially an artist, to understand the emotions and experiences of others and to sympathize with them. "The active power of empathy which makes the creative artist, or the passive power of empathy which makes the appreciator of art" (Rebecca West).
A fair-minded observer may see a great deal of overlap between sympathy as used in the quotation from Willa Cather and empathy as defined immediately afterward by Merriam-Webster.
'Sympathy' and 'empathy' in Merriam-Webster's Dictionary of Synonyms (1984)
Many of the entries in Merriam-Webster's Dictionary of Synonyms (1984) are virtually identical to their counterparts in the 1942 edition of the dictionary, aside from a smattering of replaced or additional quotations illustrating how writers use a particular word. But that is not the case with sympathy and empathy. The first change is that the synonym bundle now appears under sympathy instead of pity. But the wording of the discussions of both sympathy and empathy are completely different in 1984 from what they were in 1942:
Sympathy, pity, compassion, condolence, ruth, empathy are comparable though often not interchangeable when they mean a feeling for the suffering or distress of another. Sympathy is the most general term, ranging in meaning from friendly interest or agreement in taste or opinion to emotional identification, often accompanied by deep tenderness {sympathy with my desire to increase my ... knowledge—Fairchild} {sympathies were ... with the Roman Stoics—Ellis} {satire has its roots not in hatred but in sympathy—Perry} ... Empathy, of all the terms here discussed, has the least emotional content; it describes a gift, often a cultivated gift, for vicarious feeling, but the feeling need not be one of sorrow; thus empathy is often used as a synonym for some senses of sympathy as well as in distinction from sympathy {what he lacks is not sympathy but empathy, the ability to put himself in the other fellow's place—G. W. Johnson} Empathy is frequently employed with reference to a nonhuman object (as a literary character or an idea, culture, or work of art) {a fundamental component of the aesthetic attitude is sympathy, or—more accurately—empathy. In the presence of any work of art ... the recipient ... must surrender his independent and outstanding personality, to identify himself with the form or action presented by the artist—Read}
In my opinion, the change in MW's understanding of empathy between 1942 and 1984 reflects the professionalization of empathy in psychology and psychiatry. From being (in 1942) primarily a power of sympathetic imagination that is most common, in its active form, in creative artists, empathy becomes (in 1984) primarily a mental orientation to maximize analytical insight—a cultivated gift of vicarious feeling (one thinks of a psychotherapist trained to glean the feelings of a patient without indulging in them, or of an art appreciator passively identifying with the artist without fully internalizing the artist's emotional onslaught).
'Sympathy' and 'empathy' in Merriam-Webster Online (undated)
MW's undated online article, What's the difference between 'sympathy' and 'empathy'?, if anything, doubles down on the 1984 synonym dictionary's effort to distance empathy from a core sense of shared sympathy:
Sympathy vs. Empathy Difference
The difference in meaning is usually explained with some variation of the following: sympathy is when you share the feelings of another; empathy is when you understand the feelings of another but do not necessarily share them.
...
Empathy is Understanding [whereas 'Sympathy is Sharing']
Empathy suggests the notion of projection. You have empathy for a person when you can imagine how they might feel based on what you know about that person, despite not having those feelings explicitly communicated[.]
...
Empathy can be contrasted with sympathy in terms of a kind of remove, or emotional distance:
The act or capacity of entering into or sharing the feelings of another is known as sympathy. Empathy, on the other hand, not only is an identification of sorts but also connotes an awareness of one's separateness from the observed. One of the most difficult tasks put upon man is reflective commitment to another's problem while maintaining his own identity. —Journal of the American Medical Association, 24 May 1958
It is not by chance that the quotation that MW has chosen to highlight this proposed distinction comes from the official periodical of the AMA. It would hardly do for a clinician to dispense with emotional distance and perform some sort of Vulcan mind meld with a patient who is suffering emotional trauma or other mental turmoil.
'Empathy' in Merriam-Webster's Eleventh Collegiate Dictionary (2003)
It seems fair to ask whether everyday nonspecialists use empathy in a way that rigidly sequesters it from sympathy as the AMA recommends. My sense is that they do not. Still, Merriam-Webster's Eleventh Collegiate Dictionary (2003) studiously avoids mentioning the word sympathy in either of its two definitions of empathy:
empathy n (1850) 1 : the imaginative projection of a subjective state into an object so that the object appears to be infused with it 2 : the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings, thoughts, an experience of another of either the past or present without having the feelings, thoughts, and experience fully communicated in an objectively explicit manner; also : the capacity for this
I suspect that the vast majority of lay people do not use empathy in either of these senses. The first definition is so abstract as to be almost opaque, and the second comes across as stilted, artificial, and clinical.
'Sympathy' and 'empathy' in the American Heritage Dictionary (2010)
The entry for empathy in The American Heritage Dictionary of the English Language, fifth edition (2010) does a much better job of approximating what real-world non-experts have in mind when they use the word:
empathy n. 1. The ability to identify with or understand another's situation or feelings: Empathy is a distinctly human capability. ... 2. The attribution of one's feelings to an object: They have empathy for the evacuees who were displaced by the flood.
In a discussion of a synonym bundle that includes sympathy and empathy, AHDEL writes as follows:
pity, compassion, sympathy, empathy, commiseration, condolence. These nouns signify kindly concern aroused by the misfortune, affliction, or suffering of another. ... Sympathy denotes the act of or capacity for sharing in the sorrows or troubles of another: "They had little sympathy to spare for their unfortunate enemies" (William Hickling Prescott). Empathy is an identification with and understanding of another's situation, feelings, and motives: Having changed schools several times as a child, I feel empathy for the transfer students.
Although AHDEL supports the same basic split between "sympathy = sharing" and "empathy = understanding" that MW identifies—and although it, too, avoids using the word sympathy in its definitions of empathy—it does so in a way that permits readers to see more easily that empathy can involve a strong identification with another person's feelings—what one might be tempted to call sympathy.
Conclusion
Merriam-Webster's approach to the definitions it composes can be startlingly inconsistent. In many instances, it adopts the populist, descriptivist view that the definitions it gives for a word ought to reflect how how people use that word in the real world. But in some instances, it adopts a narrow, specialist-friendly, prescriptivist view of the proper definitions of a word, as though that word existed only in a milieu where all users were aware of and respected its precise, complicated, and nuanced technical meaning. This, I think, is what MW has done in its handling of empathy in recent decades, and as a result the meanings it endorses seem poorly matched to the ways in which people in the wild actually use it.