Short answer:

Kaiser Wilhelm probably had Erb's Palsy-- though there are other hypotheses. There was nothing wrong with Napoleon's arm.

Discussion:

Napoleon did not have a bad arm. He's frequently pictured with his hand tucked into his vest, sometimes looking like he's holding his stomach-- that can look almost like an arm in a sling. You'll hear suggestions that he had an ulcer or some other condition but that's not supported by evidence, his pose seems to have been a style.Wilhelm had what's referred to colloquially as a "withered arm", and it's said to be an injury of psychohistorical importance:

Kaiser Wilhelm II of Germany (1859) developed a weak and noticeably short left arm during childhood, commonly attributed to nerve damage caused by the use of excessive force during his difficult breech delivery, Erb's palsy. . . Wilhelm blamed the British doctor for his deformity, and formed an enmity, which ultimately led to the 1914-1918 world war.

Some physicians think that the damage could alternatively been caused by his mother's fall during pregnancy.

What's historiographically interesting about Wilhelm is that in the aftermath of World War I there was a large amount of writing attributing Germany's aggression to a psychiatric defect in the Kaiser, caused by his childhood injury. Much of it is speculative and seen from a century later, looks like the kind of thing many of of us think that we shouldn't do as historians. Perhaps the most influential of these writings was Emil Ludwig's 1926 biography, which identified his injury as the source of his belligerence.

Before the 20th century, damaged limbs were reasonably common -- either breaks that healed badly, or birth injuries. You'll often hear people saying nasty things about such people, but they're all pretty much bunk. While people understood injuries fairly well-- lots of broken legs and amputations-- they didn't understood birth injuries, and often conjectured that the damage represented some spiritual defect, or some misdeed of the mother while pregnant.

So, the French diplomat Talleyrand was born with a somewhat damaged leg and foot, perhaps a "clubfoot"-- nasty things were said about him, and people made some fun of him, but at the end of the day, all it was, was an injury. He told an unlikely story about how he came to have the injury, and there are some fairly ridiculous rumors, but it seems to have been an injury at birth. His limp did affect his life-- though the eldest in the family, his father essentially "demoted" him to second son, destined for the Church; we do have evidence from the man himself that this changed the course of his life.

And one can think of any number of other examples where some visible injury is taken to imply something about character -- Richard III's severe scoliosis became a token of his equally twisted character for Shakespeare and others.

For someone with a similar condition there's a lesson there: people have said all kinds of ridiculous things about such injuries, they can safely be ignored.

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Sources:

Jain, V., Sebire, N. J., & Talbert, D. G. (2005). Kaiser Wilhelm syndrome: Obstetric trauma or placental insult in a historical case mimicking Erb’s palsy. Medical Hypotheses, 65(1), 185–191.

Rohl JCG. Young Wilhelm: the kaiser’s early life, 1859–1888. Cambridge: Cambridge University Press; 1999. p. 1–18.

Ludwig, E. "Kaiser Wilhelm II", trans. E. C. Mayne (London, 1926)

Meyer, Arline (March 1995). "Re-dressing classical statuary: The eighteenth-century `Hand-in-Waistcoat' portrait". Art Bulletin. 77 (1): 45–63