My impression is that, in speech, 名前 is polite, and 名 is ruder - but in writing, 名 sounds more literary. Am I correct?
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2This post may answer you. – broccoli forest May 24 '15 at 01:09
2 Answers
名前 is just "name", and sometimes お名前 is used in polite conversation, but 名 would not be used in conversation. In literature you might see it like 彼の名は~~ but it sounds good only in a story context, not conversation unless you intend to sound like you are narrating something in a joking manner.
In school, when we would go to the teacher's room, sometimes teachers would tell us to 名乗れ! meaning "Say your name" if we didn't greet properly. But I have never heard 名 being used for anything else.

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Along with story context, it appears it can be used to denote reputation and level of Fame too "Name; Title; Fame; Reputation" from https://jisho.org/search/名 – greg Apr 25 '19 at 00:44
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As Haya8 said, [名前]{なまえ} means name in both written and speech.
Polite way (not used for oneself): お名前
Ex. お名前は何ですか。
Common way: 名前
Ex. 私の名前は一郎です。
名 itself isn't used in speech. However, in written terms:
It can be a counter for people. ~名
Ex. 5[名]{めい} means 5 persons.It can be a suffix of something. ~名
Ex. [会社名]{かいしゃめい} means Company Name