Do noun and name carry a different meaning?
If there are any differences, are those differences specific to a context?
I think noun is a more general concept. All the names are nouns! On the contrary, not all the nouns are names! Also, we usually talk about nouns in grammatical context.
According to Longman Dictionary of Contemporary English
Noun: a word or group of words that represent a person (such as ‘Michael’, ‘teacher’, or ‘police officer’), a place (such as ‘France’ or ‘school’), a thing or activity (such as ‘coffee’ or ‘football’), or a quality or idea (such as ‘danger’ or ‘happiness’). Nouns can be used as the subject or object of a verb (as in ‘The teacher arrived’ or ‘We like the teacher’) or as the object of a preposition (as in ‘good at football’).
Name: What someone or something is called such as 'Mandy'.
Proper noun: a noun such as ‘James’, ‘New York’, or ‘China’ that is the name of one particular thing and is written with a capital letter. It's also called proper name (especially British English).
Common Noun: in grammar, a common noun is any noun that is not the name of a particular person, place, or thing. For example, ‘book', ’sugar', and ‘stuff’ are common nouns.