I am attempting to answer this because there is not yet any answer to it, so please help me understand, if someone has a better grasp of this topic than me (which certainly includes a lot of people on JSE)
母方【ははかた】 and 母系【ぼけい】 - both are synonyms for "maternal" just as
父方【ちちかた】 and 父系【ふけい】 are synonyms for "paternal"
However 母方【ははかた】 and 父方【ちちかた】 have the literal meaning of "on my mother's/father's side", while
母系【ぼけい】and 父系【ふけい】 have the literal meaning of "in my mother's/father's (familial) group".
For the purposes of discussing a person's individual relatives, for example, "my grandfather on my father's side" or "my paternal grandfather", there is little semantic difference. You can say either 父方【ちちかた】の祖父【そふ】 or 父系【ふけい】の祖父【そふ】
For other topics though, such as "patriarchy" or "matriliny", or other such more complex concepts, different concepts require one version over the other, for example: "matriarchy" = 母系制 【ぼけいせい】 and "partilocal" = 父方居住【ちちかたきょじゅう】
There may also be issues where "mothers-in-law" or "fathers-in-law" and members of their family/issues relating to them require special wording, of which I am unaware.
To go into why one form is used over the other form for complex concepts, is beyond my level of Japanese proficiency. Hopefully a linguist will pick up where I can no longer contribute.