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Disaster, catastrophe, and calamity all seem to have very similar, if not the same, definitions in dictionaries. So, how do they, if at all, differ from each other?

Here is how Merriam-Webster's Dictionary of Synonyms (1984) distinguishes the three words from one another:

disaster, calamity, catastrophe, cataclysm are comparable when they denote an event or situation that is regarded as a terrible misfortune. A disaster is an unforeseen mischance or misadventure (as a shipwreck, a serious railroad accident, or the failure of a great enterprise) which happens either through culpable lack of foresight or through adverse external agency and brings with it destruction (as of life and property) or ruin (as of projects, careers, or great hopes). [example omitted] Calamity is a grievous misfortune, particularly one which involves a great or far-reaching personal or public loss or which produces profound, often widespread distress; thus the rout at Bull Run was a disaster for the North but the assassination of Lincoln was a calamity; the wreck of the Don Juan was a disaster and, as involving the loss of Shelley, it was a calamity [further examples omitted] Catastrophe is used of a disastrous conclusion; it often emphasizes the idea of finality [examples omitted]

Although the 1984 publication date associated with this book may make it seem to be a fairly recent assessment of how the three words differ, the account quoted above is taken almost verbatim from Webster's Dictionary of Synonyms (1942), so the treatment is now almost 75 years old.

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) discusses the same three terms as part of a group of similar terms that also includes cataclysm and debacle. Here is Hayakawa's discussion of the group as a whole and of the three relevant words:

These words refer to misfortunes that result in grave loss or heavy casualties. Catastrophe is equally appropriate for a personal or public misfortune: [examples omitted]. In personal application, the word is often used hyperbolically of minor incidents: [example omitted]. In reference to a general event, the word may refer to the negative effect on a particular group rather than to the public as a whole: [example omitted]. ...

Disaster is the most general of these words, referring both to personal and public misfortunes in a wide range of possibilities: [examples omitted]. Like cataclysm, the word can refer to natural upheavals, but without the implications of total destruction present in cataclysm: [example omitted]. Disaster compares with catastrophe by stressing the actual harm done: [example omitted]. The word can, of course, be used hyperbolically, like catastrophe, for minor misfortunes [example omitted].

Calamity is similar to catastrophe, but at a reduced level of intensity. It may now sound more formal than these other words, or at least a shade outdated. It is more often used for a personal misfortune, seriously or hyperbolically, but it can also be used of public misfortunes on occasion, in which case it functions more abstractly or subjectively than disaster; [examples omitted].

James Fernald, English Synonyms, Antonyms, and Prepositions (1899) offers a view of how the words seemed to differ more than a century ago:

A cataclysm or catastrophe is some great convulsion or momentous event that may or may not be a cause of misery to man. In calamity, or disaster, the thought of human suffering is always present. It has been held by many geologists that numerous catastrophes or cataclysms antedated the existence of man. In literature, the final event of a drama is the catastrophe, or denouement. ... In history the end of every great war or the fall of a nation is a catastrophe, tho it may not be a calamity. Yet such an event, if not a calamity to the race, will always involve much individual disaster and misfortune. Pestilence is a calamity; a defeat in battle, a shipwreck, or a failure in business is a disaster; sickness or loss of property is a misfortune; failure to meet a friend is a mischance; the breaking of a teacup is a mishap.

One of the most interesting points that Fernald makes is that in 1899 catastrophe had a strong connection to a dramatic finale. That overtone of the word has disappeared in the modern day. However, the use of catastrophe (and cataclysm) to describe events that occurred before the evolution of human beings seems to have persisted. The notion that a calamity may be more personal than a disaster or a catastrophe is plausible and may be true in some situations today.

Also of possible interest are the usage frequencies of disaster (blue line), catastrophe (red line), and calamity (green line) over the period 1800–2003, as reflected in this Ngram chart:

In 1800, calamity was far more common in published English texts than either of the other two words. But since about 1880, disaster has been the most common term of the three; and since about 1920, catastrophe has bypassed calamity as well. It seems very possible that at least some of this shift in usage frequencies reflects the fact that people sometimes use disaster today to describe what they would have called a calamity 200 years ago.

Disaster: unfortunate event causing destruction.

Catastrophe: sudden disaster.

Calamity: disaster with long-lasting effects.