Is "entity" just a fancy synonym for "thing"?

Are there any noteworthy differences between the meanings of these words? Is "entity" just a fancy synonym for "thing"?

The definitions for "entity" and "thing" are referencing each other.

You may be surprised to learn that Merriam-Webster's Dictionary of Synonyms (1984) doesn't group entity and thing in the same group of words at all. Instead entity is the lead term in a group that also includes being, creature, individual, and person, while thing is the lead term in a group that also includes object and article. The dictionary explains these groupings as follows:

entity, being, creature, individual, person are comparable when meaning something which has real and independent existence.

...

Thing, object article are comprehensive terms applicable to whatever is apprehended as having actual, distinct, and demonstrable existence. They vary, however in their range of application.

Hmmm. Those groups don't sound so very different after all, do they? Okay, then, let's look at the dictionary's discussion of the specific words entity and thing:

Entity, the most consistently abstract of these terms [in its group], implies such existence not only in the actual world but also in the realm of thought. An entity may be be seen or heard or it may be invisible, intangible, or imaginary, but it may be thought of as really existing {that entity which we call an automobile} {is democracy an entity?} {his country is to him an entity, a concrete and organic force, with whose work in the world he is extremely proud to be natively associated—Brownell} (I introduce the entity called light to the readers of this book: as an entity consisting of particles—Darrow} {for the good of that mystical entity, different from and superior to the mere individuals composing it, the Nation—Huxley}

...

Thing is the term of widest reference [in its group]. In its most inclusive sense it need not imply direct knowledge through the senses but is equally applicable to something so known and to something the existence of which is inferred from its signs or its effects; thus, one thinks of the state, the church, literature, and the law as things rather than as ideas or abstractions; a friend's affection is as real a thing as as is his house or his hand; one distinguishes a word from the thing it names {name the things that are on this table} {wanted to do the right thing} {a blind person recognizes things through such qualities as shape, texture, smell, taste, and sound} In somewhat more restricted use thing can denote specifically an entity having existence in space or time as distinguished from one existing only in thought {virtue is not a thing but an attribute of a thing} or in still more restricted use an inanimate entity and especially a material possession as distinguished from living beings and especially persons {more interested in things than in human beings} Often the word is used idiomatically to mention without specifically identifying an item that cannot or need not be further identified or whose nature is implicit in the context; thus, in "be sure to wear warm things," clothing is implied; in "bring in the tea things," the necessary collection of dishes, implements, and foods is implied {what's that thing in your hand?} Occasionally thing may be used in reference to persons when contempt is expressed or derogation intended {do you call that thing a man?}

These descriptions indicate considerable overlap between the two words' meanings. But the included examples show something else—that though thing might be able to replace entity in every instance of the latter word ("that thing called an automobile," "is democracy a thing," "his country is to him a thing," "the thing called light ... as a thing consisting of particles," "that mystical thing ... the Nation"), the reverse is not true.

"A friend's affection is as real an entity as is his house or his hand" and "virtue is not an entity but an attribute of an entity" may pass muster, but others clearly do not: "do the right entity"; "be sure to wear warm entities"; "bring in the tea entities." Even "more interested in entities than in human beings" sounds a bit off, perhaps because we can more easily recognize the opposition of things to human beings than of entities to human beings.

Harry Shaw, Dictionary of Problem Words and Expressions (1975) offers this interesting comment on the word thing:

thing. This is an all-purpose word used so loosely that it often has no real meaning. For instance, instead of "One thing I like about him ..." why not say "one characteristic," "one trait," or "one distinctive feature"? Because thing means "whatever can be thought or believed to have an existence," a good rule to follow is this: one should never say thing unless he has some specific entity (object) in mind and then mentions the entity itself. This is a thing (rule, prescription, item of advice) none of us will ever achieve or do, but it's a thing (endeavor, activity, counsel of perfection) we should try to follow.

What makes this comment especially useful in the context of the poster's question is that—even though Shaw enjoins readers to find the specific entity behind the general term thing and to use that more specific term instead—not one of the phrases he cites in his question works if you swap out thing in favor of entity. Consider: "One entity I like about him"; "This is an entity none of us will ever achieve or do"; "it's an entity we should try to follow."

In other words, linguistically, entity often behaves as if it were a sort of anti-thing: We might readily use the categorical term entities in a sentence that talks about "entities such as characteristics, traits, features, rules, prescriptions, items of advice, endeavors, activities, or counsels," but we would almost certainly resist replacing any of those particular so-called entities with the word entities. In sharp contrast, when it comes to the word things, we happily abandon the more specific in favor of the more general—which is the source of Shaw's complaint about thing.

To sum up, in abstract or technical settings, entity might with some justice be called "just a fancy synonym for thing." But the more narrowly a particular category of thing is implied—and especially the more figuratively or idiomatically thing is used—the less likely entity is to be able to stand in for thing and still sound like normal English.