Is there any difference between the words (nouns)
anguish? They seem to be quite similar.
Webster's Dictionary of Synonyms (1942) assigns the two words to different groups of related terms. Anguish falls in the sorrow category (along with woe, heartache, grief, dole, and regret), while distress is the lead term in a group that includes suffering, misery, agony, dolor, and passion. This dictionary has this to say about the special features of the two words:
Anguish is excruciating or torturing grief or dread."Anguish so great that human nature is driven by it from cover to cover, seeking refuge and finding none" (R. Macaulay). "I had that terrible pain before playing—that anguish which is not to be described" (Paderewski).
Distress, in precise use, commonly implies conditions that cause physical or mental stress or strain; usually also it connotes the possibility of relief or the need of assistance. [examples omitted]. The word is applicable to things as well as to persons; thus, a ship in distress is helpless and in peril because of some untoward circumstance such as a breakdown in machinery or the loss of necessary equipment; a community's distress may be the result of a disaster or any event bringing devastation with it or imposing extreme hardship on the people. When used to designate a mental state, distress usually implies the stress or strain of fear, anxiety, shame, or the line. [Examples omitted.]
James Fernald, Funk & Wagnalls Standard Handbook of Synonyms, Antonyms & Prepositions (1947) lists anguish in three categories of related terms (anxiety, grief, and pain), and distress in four categories (grief, misfortune, pain, and poverty). In the entry for the shared category of pain, Fernald distinguishes between the two terms in this way:
Distress is too strong a word for little hurts, too feeble for the intensest suffering, but commonly applied to to some continuous or prolonged trouble or need; as, the distress of a shipwrecked crew, or of a destitute family. ... Agony and anguish represent the utmost pain or suffering of body or mind. Agony of body is that with which the the system struggles; anguish, that by which it is crushed.
S. I. Hayakawa, Choose the Right Word (1968) lists anguish under misery, and distress under both misery and hurt. Here are the relevant parts of the coverage of misery and related terms:
Distress is too strong a word for little hurts, too feeble for the most intense suffering. It more often than not is applied to mental states, referring to any deep anxiety or the external circumstances that may produce it. Very commonly it applies to some prolonged trouble, as does misery, but distress more than misery implies at least a possibility of relief: the distress of an underprivileged child. ... Anguish points to the extremity of grief which so terrifies the spirit as to be insupportable: the anguish she knew when when her husband and three children were burned to death, anguish so great that it turned into madness.
So it appears that anguish generally refers to a more extreme and intense level of suffering than distress does, and also one that doesn't hold out as much prospect of relief by someone offering aid. That is, a damsel in distress may be restored to safety and tranquility by a knight errant, but a damsel in anguish may, perhaps, not be.
(Incidentally, I don't think that the repetition in Hayakawa of Fernald's sentence about "little hurts" is an instance of plagiarism, since the Hayakawa book was originally published by Funk & Wagnalls and was, I believe, conceived as a major updating and modernization of Fernald.)
In those senses of each which are most nearly synonymous, distress differs from anguish chiefly in more readily admitting of a minor degree, i.e., being modifiable by such attenuators as slight and little: ngram.