That's a very interesting question. It is all in the connotation.
In U.S. English, "farmer" is the usual word. It is neutral, factual, polite, and refers to everyone who farms--large or small, rich or poor.
"Peasant" literally means a small farmer. However, in U.S. English it tends to be used about a) European farmers b) in the past. And it is definitely a negative word. It has overtones of being poor, crude, rustic, and unsophisticated.
In the United States we speak of "small farmers," "tenant farmers," "poor farmers," and "sharecroppers." These fall within the meaning of "peasants" but we would never say "peasants." We do not have "peasants" in the U.S.!