Do the words "peasant" and "pissant" mean the same thing?

A peasant is a farmer. A "pissant" is someone who throws his words away like "water." In practice, handworking peasants are seldom "pissants." Their only similarity is that the words can sound alike if not pronounced distinctly.

I think you've got your words confused. Here is a wiki entry on the word. Quoted from Wikipedia:

A pissant, also seen as piss-ant and piss ant, is one or the other of two specific types of ant. Its origin is with pismire, a 14th-century word for ant. The term is also used as an insulting noun, and a pejorative adjective.

OK, several separate matters are at issue here.

First, the word Vonnegut describes is pedant; however, the mix-up was likely intentional – a means to reveal a certain confusion in his character’s state of mind. (This rhetorical device has a name; it’s not quite occultatio, but is related to the idea of a “Freudian slip” – alas, I forget).

Some say peasant = “farmer”. Well, so is “agronomist” – denotatively. In connotation, “peasant” especially emphasizes the insignificance of poor peons as might “come with the land” (paisano = “countryman”) in a rural country.

Wikipedia, basking in its undeserved certitude, often blinds itself (and others) with false detours. Its tangent about actual ants is irrelevant and unrelated. In AmE, the noun pissant derives from the adjective; both are one word. One might say, “that pissant (adj.) guy said …” or for short, “that pissant (n.) said …” Either way, the point is insignificance, which it inherits – consciously or not (like “cusses” from “curses”) – from peasant.

Never attribute to erudition what can be adequately explained by malapropism.

Pissant is a kind of ant but it also a pejorative term on piss (urine) and peasant and partly a joke on some American pronunciation of peasant.

I always thought it was "piss ant". It's reasonably common slang, at least among Americans of a certain age. It's definitely not the same word as "peasant"; the Wikipedia entry makes no mention of peasants and has a plausible etymology having to do with ants that smell like urine.