Difference between Peasant and Farmer

That is misconception. Famer is not the same thing as peasant.

Peasant works on land that is owned by landlord (aristocrat) and either pay him rent for using his land or does some work for him in lieu of rent. Peasant can feel discontent with working for rich fat asshole who does nothing in peasant's opinion yet profits from peasant's work and can afford luxury lifestyle while peasant scarps by.

Farmer on the other hand owns the land he works on, kind of like a free peasant but with much more attitude: that is his land and his farm and he does whatever the fuck he pleases on it. The only thing farmer hates is to pay taxes from money he makes selling his harvest every year.

Peasant is proletariat and farmer is petit bourgeoisie in Marx terms.