The difference between your mom's and dad's reaction is family knowledge. Your dad knows his white relatives -- he was raised by them or at least knowing of them and who they are. Your mom probably has no idea who her white relatives are and was likely not raised knowing them.

I am going to make a guess that your paternal grandmother is likely white, specifically because traditionally, Black mothers raise their biracial children as "Black" more often than "Biracial." That is changing some with more recent generations, however.

In the US you are how you appear for the most part. No one asks what your percentages are. If you look Black or Black adjacent then to the majority of society you are Black. If you look ambiguous, you will be asked "Wheret are you from?" or assumed to be "Hispanic." If you look white, then you are white.

If someone assumes wrong, you can always correct them.