I'm also curious about 祖母 vs 奶奶 , 父亲 vs 爸爸 ， and other family words.
In the context of standard Mandarin, the main difference is that the former one listed in all your three groups are used in more formal situations while the latter ones are more colloquial.
Besides those you have mentioned, in colloquial Mandarin it is also common to use monosyllable words when talking with family members or referring to family members. Such as 爸，我回来了。and 我爸去年刚退休。Monosyllable words like 父 and 母, on the other hand, are usually used to form compound words like 父母, 生母, 养母.
There are other words that refer to family members that sounds old-fashioned and are used as the honorifics. For example, 令尊 is "your father" and 令堂 are "your father" and "your mother" with honorific tones, respectively. It is not suggested to use these unless you are sure that it is the appropriate situation to use them.
祖父 父亲 is like grandfather, father, while 爷爷 爸爸 is like grandpa, dad. I think.
祖父 and 祖母 are old-fashioned and they are usually used in written to tell a old story in the old days.
爸爸, 爷爷 and 奶奶 are frequently used in our daily life. The single character 奶, 爷 or 爸 is common in more casual situations.
Well, 父亲 and 母亲 are usually used to formally clarify the father or mother relationship. For example, 这是我的父亲/母亲。You are trying to introduce your father or mother formally, indicating father-and-son or mother-and-son relationship.
Side note in addition to other answers: Beware in the Cantonese context, 奶奶 does not mean 祖母 but rather the mother-in-law of the wife.