Rude is an adjective in English, describing someone or an action that is considered unpolite. For example, "it is rude to not say thank you when receiving a gift".
Rough (pronouned like ruff) has many meanings.
Firstly, it can be used as an adjective to describe a surface or terrain. It is the opposite of something being smooth, as in it has no lumps or bumps. Examples of things that are rough are sandpaper, or the sea when there are many waves.
It can also be used tp describe an experience or trip that was not easy.
Other meanings can relate to children when thay play tumble with each other that is risking one person getting hurt however not as often.
The range of meanings overlaps a lot, more than some people might think.
As others have noted, "rude" usually refers to bad behavior. Often, it means bad in the sense of vulgarity. For example, spitting in public is rude. Using bathroom-related bad language is rude. It can mean simply disrespectful or impolite.
"Rough" can mean the opposite of "smooth;" "having an irregular surface," unpolished. That's the most common meaning. "This board was rough. I used sandpaper to smooth it." "This is just a rough draft of the essay, I need to polish it." It can also mean a difficult situation. "He still hasn't found work yet. He's having a rough time."
When it refers to behavior, there is usually an implication of violence. "Roughed up" means assaulted, hit, beaten. "Roughs" or "Rough men" are violent men. There is a quotation, often attributed to George Orwell: "We sleep soundly in our beds because rough men stand ready in the night to visit violence on those who would do us harm."
"Rude" can apply to physical objects and have about the same meaning as "rough" or "crude:" not finished, not refined. This is an old-fashioned meaning.
A poem famous in the US, written in 1837, commemorates the first battle of the American Revolution in 1775. It begins:
By the rude bridge that arched the flood,
Their flag to April’s breeze unfurled,
Here once the embattled farmers stood,
And fired the shot heard round the world.
The bridge didn't smell bad or behave disrespectfully. "Rude" means that it was built simply. It was probably made of rough timbers. It wasn't a beautiful piece of construction. This is a modern reconstruction; it is "a rude bridge."