The reason "Mom" is capitalized in the first sentence is that "Mom" is being used as an address. It's standing in for her name, and therefore takes on the function of a proper noun.
In the sentence "She became a mom", "mom" functions as a standard noun, so it's lower case.
The following excerpt from New Hart's Rules, section 5.10 (p. 97) explains the rule clearly:
Words indicating family relationships are lower case unless used as part of a name or in an address.
It gives the following examples, which should help with understanding exactly when words such as "mom" should and should not be capitalized:
| Not capitalized |
Capitalized |
| he did not look like his father |
Hello, Father! |
| she has to help her mother |
Maya tried to argue with Mother |
| ask your uncle |
Uncle Brian |