difference between coward and pusillanimous

The two words have similar meanings, but different implications. Perhaps most significant is that cowardly tends to imply something about the subject's actions, while pusillanimous bears more on their personality and behavior.

A pusillanimous person would tend to not take a position in a political discussion, eg, while a cowardly person might denounce an autocrat in private but would be loath to denounce the autocrat to his face.

Although there is indeed a considerable overlap between the two, they are not quite the same. Pusillanimous can be used as a rough synonym for timid, while cowardly cannot.

Cowardice always involves fear of some relatively definite adverse consequences. A soldier may be cowardly in a battle because he fears being killed by the enemy. A cowardly employee never contradicts his boss, because he fears that the boss may retaliate in some way.

A pusillanimous person, on the other hand, need not have any such reason for behaving in pusillanimous ways. A pusillanimous person will not contradict his boss even he is quite confident that the boss won't retaliate. It's just in his nature that he doesn't contradict people; he is not assertive enough for that, and he doesn't like drawing attention to himself.

Webster's Dictionary of Synonyms, first edition (1942) addresses cowardly and pusillanimous in an entry that also includes the allied terms poltroon, craven, dastardly, and recreant:

Cowardly, pusillanimous, poltroon, craven, dastardly, recreant agree in meaning excessively timid or timorous. Cowardly, the most general term, implies a weak or ignoble, pusillanimous a mean-spirited and contemptible, lack of courage; as, "He...plac'd behind With purpose to relieve and follow them, Cowardly fled, not having struck one stroke" (Shak[espeare]); "Cowardly dogs! ye will not aid me then?" (Shelley); "I lived in a continual, indefinite pining fear; tremulous, pusillanimous, apprehensive of I knew not what (Carlyle); "having no materialized class above it, it {American vulgarity} is not obsequious and pusillanimous" (Brownell).

Interestingly, Merriam-Webster's Dictionary of Synonyms (1984), the successor to the first edition of Webster's Dictionary of Synonyms, drops all coverage of cowardly, pusillanimous, craven, dastardly, and the rest. The omission is all the more inexplicable in view of the fact that Merriam-Webster's Ninth Collegiate Dictionary (1983) includes the following usage note at the end of its entry for cowardly:

COWARDLY, PUSILLANIMOUS, CRAVEN, DASTARDLY mean having or showing a lack of courage. COWARDLY implies a weak or ignoble lack of courage; PUSILLANIMOUS suggests a contemptible lack of courage; ...

In its usage note, the Eleventh Collegiate Dictionary (2003)—the most recent in the series—adds examples of cowardly ("a cowardly failure to stand up for principle") and pusillanimous ("the pusillanimous fear of a future full of possibility") but otherwise retains the same description of each that appears in the Ninth Collegiate. So either the lexicographers at Merriam-Webster consider the difference between "a weak or ignoble lack of courage" and "a contemptible lack of courage" to be fundamental and obvious or they haven't bothered to revisit the distinction since they set it down back in 1942, even as the Dictionary of Synonyms branch of the organization has withdrawn from the discussion altogether.

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) offers this commentary on cowardly and pusillanimous, which it discusses along with craven:

cowardly craven pusillanimous These words mean lacking courage to a degree that arouses disapproval and disgust. Cowardly is the most common word of the three, and is applied opprobriously to persons who are unwilling or unable to prevent their fear or timidity from influencing their actions unduly; it can also refer to the actions themselves: In frontier days, shooting a man in the back was considered cowardly; too cowardly to stand up and fight.

Craven and pusillanimous are formal words encountered mainly in writing. ... Pusillanimous differs from the other terms in in pointing more strongly to temperamental timidity than to fear as the basis of the resulting action or inaction. Pusillanimous represents a contemptible moral squeamishness rather than a physical cowardliness, although it is quite possible for the same person to be both pusillanimous and cowardly. What chiefly distinguishes the pusillanimous person, however, is his unwillingness to press for his rights: His pusillanimous reaction was to sigh and say, "Well, it really won't do to raise a fuss."

These various attempts to distinguish between cowardly and pusillanimous touch upon several possible points of difference:

1. Cowardly is a more general term and is therefore applicable to a broader range of situations.

2. Pusillanimous is a more literary term and is therefore less likely to be encountered on the streets of Laredo.

3. There is some sense that cowardly may apply to situations where the act (or failure to act) is understandable, though it fails to meet even a minimum standard of appropriate conduct under the circumstances; that is, in some cases, it may apply to an uncharacteristic but explicable failure of nerve and thus indicate a temporary shortcoming. Pusillanimous, on the other hand, seems to go to a habit of mind—a fundamental, ever-present lack of bravery that leads to cowardly behavior even when meeting an acceptable standard of conduct wouldn't require unusual fortitude or high-mindedness; this would explain the association (by both Merriam-Webster and Hayakawa) of contemptible with pusillanimous but not (or not necessarily) with cowardly.

The main difference between the terms coward and pusillanimous is one of usage, not in meaning. The overwhelming majority of native speakers will use "coward/cowardly/cowardice" exclusively in speech, and have only read or heard of "pusillanimous" while studying for the SAT or GRE test.

Etymonline says
early 15c., from Late Latin pusillanimis "having little courage" (used in Church Latin to translate Greek oligopsychos "small-souled"), from Latin pusillis "very weak, little" (diminutive of pullus "young animal," from PIE root *pau- "few, little") + animus "spirit, courage".

Which literally means the “small courage of a soul”.

Fans of the American TV show "The Sopranos" will remember a mob gangster whose nickname was Big Pussy, which could be highly offensive (slang for vagina) or insulting, “You must do this, you big pussy!” (whether the cowardly meaning of pussy is the abbreviation of pusillanimous is debatable but it cannot be excluded) instead, according to Wikipedia, the character's name refers to the feline variety.