What is the difference between "hurry" and "rush"?

Merriam-Webster's Dictionary of Synonyms (1984) puts hurry as a verb and rush as a verb in separate synonym clusters. Hurry appears in a group with speed, accelerate, quicken, hasten, and precipitate:

Speed, accelerate, quicken, hasten, hurry, precipitate can mean to go or make go fast or faster. ... Hurry implies haste that causes confusion or prevents concentrated attention {his aim was hurried and his shot went wide of the mark} {a second fear ... which madly hurries her she knows not whither—Shak.} {these defects might pass more easily in a turbulent romanticist, hurrying pell-mell to get expressed some moving and dramatic scene, careless of details—Fry}

Meanwhile, rush appears in a group with dash, tear, shoot, and charge:

rush, dash, tear, shoot, charge can all mean to move or cause to move forward with speed. Rush suggests either impetuosity or intense hurry on account of some exigency, and often carelessness about the concomitant effects of the precipitate action {rush for a train} {rush a research paper into print} { flying rout of suns and galaxies, rushing away from the solar system—Forster} {business rushed forward into the glittering years—Amer. Guide Series: Ind.}

Because MW doesn't compare hurry and rush side by side, the only suggestion of difference I can glean from the parallel discussions of the two terms is the idea that rush represents an intensified form of hurry. But this inference arises only obliquely, and ultimately the Merriam-Webster treatment of the two terms isn't terribly useful. I note, too, that the predecessor to this dictionary, Webster's Dictionary of Synonyms (1942), provided no coverage of rush at all. This omission might point to increased use of rush in the sense of "hurry" in the decades since 1942, or it might indicate a rather scattershot approach to identifying and distinguishing between synonyms in that earlier volume. But even the newer MW Dictionary of Synonyms is only marginally helpful in clarifying how hurry and rush differ.

A more direct comparison of hurry and rush as verbs appears in S.I. Hayakawa, Modern Guide to Synonyms and Related Words (1968), which bundles both words in a group of synonyms that also includes quicken, accelerate, expedite, hasten, and speed:

quicken, accelerate, expedite, hasten, hurry, rush, speed These words mean to move or cause to move faster. ... Hurry and rush are similar to hasten, but suggest in addition precipitous or confused motion. {The late arrivals were hurried to their seats; The stricken man was rushed to a hospital.} Rush suggests greater urgency than hurry, and sometimes includes the notion of violent action: They suddenly rushed pell-mell out the door.

I think that the distinction in Hayakawa's examples between "hurried to their seats" and "rushed to a hospital" does a good job of distinguishing between the degrees of urgency that (often) are involved in actual usage of the two terms. It's not that no one would ever say "they were rushed to their seats" or "he was hurried to a hospital"—but the word choices that Hayakawa cites strike the right level of intensity for the contexts in which they appear. His view that rush typically implies greater urgency or even violence than hurry is, I think, correct more often than not.