I saw these words a while ago but just now I decided to ask it here because I didn't find the answer anywhere.
As the title says, I don't really know if there is a difference between 母方【ははかた】 X 母系【ぼけい】, and 父方【ちちかた】 X 父系【ふけい】as well.
The sentences I've come across are:
父方【ちちかた】の祖父【そふ】の米寿【べいじゅ】のお祝【いわ】いにマフラーを贈【おく】った。
and
母系【ぼけい】の祖母【そぼ】はその旧式【きゅうしき】のドレスを着る【きる】ことに固執【こしつ】する。
First, let me point out that your second example,
母系の祖母はその旧式のドレスを着ることに固執する。
is a very unnatural sentence. You could hardly say something like that, if not never. When I searched 母系の祖母 I only found this example from Tanaka Corpus, which contains many errors and should not always be deemed as authoritative Japanese usage (see their disclaimer).
Instead you should always say:
父方の祖父 paternal grandfather
母方の祖母 maternal grandmother
Generally speaking, 母方／父方 is what you call your mother/father's house or kinship from your perspective, so you are mentally facing your parents' direction. 母系／父系 means matriline/patriline, which is a line of descent passed down from your ancestors to you (they are genealogical terms). For this reason, 母系／父系 is usually used to describe hereditary, or inheritance matters, that conceptualized in descending order, not when you call your grandparents.
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.
