What situations would one use 祖父 as opposed to 爷爷? 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.