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Similarities: they both try to prove a point, both persuasive
Differences:

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In a letter you state, who the letter is for.

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they both have a point to get
across

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A letter can contain more of personal experience or belief while trying to get your point across while the paper should be more formal excluding personal opinion and getting a point across.

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They can both have ethos, logos, and pathos

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Similarities: They are both trying to prove a point. They use rhetoric.
Differences: The letter is to one person, the paper is towards everyone. In a letter you include things that are geared towards the person you are writing to. They have different structures.

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an argumentative letter is more like direct and its structured differently it's more into evidence and rebuttals and etc. and in a letter is  towards a person and it's basically your words like your opinion

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In an angrument letter, you have to have evidence, rebuttas and support.
Agrumentative paper claim based on opinion, audience, persuasive

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I personally feel like an argumentative letter is about an argument in your POV and ur personal beliefs but you have a think about the audience while writing the letter and a Argumentative paper it has to be formal and you have to MLA cite it.

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There both argumentative
there both factual
the letter can be shorter
a paper tends to be longer
the letter is more directed

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An argumentative letter doesn't is a lot less formal than an argumentative paper. You do not need citations in a letter, while you do need it in a paper. They both will try and persuade you. Who you're talking to changes because a letter is directed to a person while a paper is directed to a bigger audience.

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A letter is towards one person, a paper can be read to a larger audience.