Near the end of I Samuel 27, I see “我们 袭击 了 犹大 的 南方， 耶拉蔑 的 南部， 和 基尼 的 南方。”
macOS built-in dictionary gives
Are they the same thing, merely used to avoid repetition, or is there actually a significant difference of meaning?
南方 = South side (inside or outside of the border)
Toronto is in Canada's 南方
Canada's 南方 is the United States of America
南部 = Southern part (within the border)
Toronto is in Canada's 南部
New York City is in New York State's 南部
方 can simply mean any area (but tends to be used to mean a region that's more general and possibly somewhat vague/ill-defined).
部 means more specifically a part/section/subset of a larger area (usually more precisely defined).
So to refer to "the West" (Europe and its offshoots like the US), we'd use 西方 but not 西部.
Similarly, in the context of China, 南方 refers to the somewhat vague and ill-defined "South" (where most speakers of non-Mandarin Chinese languages live).
There's broad agreement that Guangzhou and Fujian are in 南方 and Beijing is in 北方, but much less agreement on whether cities/regions such as Shanghai or Nanjing should be considered 南方 or 北方. So, using "方" makes sense (these are vague and imprecise concepts). Using "部" instead would incorrectly suggest a greater degree of clarity and precision than exists.
These are usually translated to Chinese as 美国南部, 美国西部, 美国东北部, and 美国中西部. We could also have used "方" for each of these four regions, but "部" works well because it tells us that these are fairly well-defined parts/sections/subsets of a larger area (the US).
Here's the text from the CNVS translation you gave:
亚吉问：“你们今天袭击了甚么地方呢？”大卫回答：“我们袭击了犹大的南方，耶拉蔑的南部，和基尼的南方。”
As Tang Ho has already pointed out, "耶拉蔑的南部" is unambiguous and means some southern part of 耶拉蔑.
In contrast, 犹大的南方 is ambiguous (in line with my above remarks about 方 being sometimes vague and imprecise): It could mean some (1) some southern part of 犹大; (2) some area south of and entirely outside of 犹大; or (3) some combination of 1 and 2.
The original Hebrew seems to be this:
,י וַיֹּאמֶר אָכִישׁ, אַל-פְּשַׁטְתֶּם הַיּוֹם; וַיֹּאמֶר דָּוִד, עַל-נֶגֶב יְהוּדָה וְעַל-נֶגֶב הַיְּרַחְמְאֵלִי
.וְאֶל-נֶגֶב, הַקֵּינִי
I don't know Hebrew so here are some English versions of the same sentences:
And Achish said, Whither have ye made a road to day? And David said, Against the south of Judah, and against the south of the Jerahmeelites, and against the south of the Kenites.
When Achish asked, “Where did you go raiding today?” David would say, “Against the Negev of Judah” or “Against the Negev of Jerahmeel” or “Against the Negev of the Kenites.”
(Note: Negev also meant "south".)
“Where did you make your raid today?” Achish would ask.
And David would reply, “Against the south of Judah, the Jerahmeelites, and the Kenites.”
Note that similar to the Chinese "的南方", the English "the south of" (or "the Negev of") is similarly ambiguous and could mean any of possibilities 1, 2, or 3 just given above.
Each of the above English translations seems to use the exact same phrasing for each, so your chosen Chinese translation seems to be simply engaging in some unnecessary and confusing elegant variation by switching from 南方 to 南部 and then back again to 南方.
Indeed, all of the other Chinese translations (at the link you gave) do not commit this error (and stick to the same consistent phrasing for each of Judah, Jerahmeel, and Kenites):
CCB:
当亚吉问：“你们今天攻打了哪里？”大卫会说：“攻打了犹大南部”，或说：“攻打了耶拉篾南部”，或说：“攻打了基尼南部。”
CSBS:
每当亚吉问：“你们今天侵袭了哪里？”
大卫就会回答：“犹大的南地”，或是“耶拉篾人的南地”，或是“基尼人的南地”。
CUNPSS-上帝 (and the same for CUNPSS-神):
亚吉说：「你们今日侵夺了什么地方呢？」大卫说：「侵夺了犹大的南方、耶拉篾的南方、基尼的南方。」
RCUVSS:
亚吉说：“今日你们没有去抢夺什么地方吧？”大卫说：“侵夺了犹大、耶拉篾、基尼等地的南方。”
普通话：
每当亚吉王问: “这次你上哪儿打仗去了？”大卫就回答说“上犹大南部去了”，或“上耶拉篾的南部去了”，或“上基尼人的南部去了”。
