In terms of meaning, what is the difference (if there is one) between these sentences:
Grandmother buys a new belt.
The grandmother buys a new belt.
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In the first sentence (assuming the omission of the article is not a mistake), I can only assume that Grandmother is being used as if it were a proper noun. In other words, that's what her family members call her. – Jason Bassford Jan 19 at 10:35
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Grandmother there is a form of address, and is used like a name.
The grandmother refers to her position or role in the family.
Compare this caption for a hypothetical photograph:
Sis gets her hair styled.
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"Grandmother" as a proper noun (similar to a name) suggests that it's the speaker's grandmother or a well-known grandmother character in a story, and furthermore that there is only one grandmother in question. If there are multiple grandmothers one would refer to them with additional name, like "Grandma Jill", or "grandmother Jones".
Today we all went to the mall, and grandmother bought a new belt.
"The grandmother" uses "grandmother" as a noun. It can be used to refer to a person who is a grandmother and has been already mentioned before, but is not a significant character, or nor familiar enough to be called "grandmother" as a name.
A girl and her grandmother went to the mall. The grandmother bought a new belt.
