Are there particular rules on when to use one or the other? 休〜 appears to be
more of a temporary closure or suspension (with implied continuation at some
point in the near future)...

>   * この図書館は改装のため、来月まで休館となっています。ご了承ください。 → Due to our remodeling, this library
> will be closed until next month. Thank you for your understanding.
>   * あしたは休講です。 → There is no class tomorrow
>   * _はしか_ の発生により、来週は休校です。 → Due to a measles outbreak, (this) school will be
> closed next week. → I actually saw this when I lived in Japan.
>

...While 閉〜 is more like a complete shutdown of something.

>   * 閉校 = [廃校]{はい・こう} → Close down a school
>   * 閉業 → Close down a business
>   * 閉場 → Close a theatre/meeting room
>

However, there seems to be some overlap in some terms.

>   * 閉店 = 休業 → Close a store for the day (to open again the next day)
>     * Whereas 休店 seems to be used when a store will be closed for an
> extended period (owner going out of town, etc.)
>   * 閉館 → Close (down) a 館
>     * 図書館・美術館など「館」と名のつく施設などが，その日の業務を終えること。 OR
>     * 図書館・映画館などがその業務をやめて施設を閉鎖すること
>     * _(Definitions taken from whatever version of **スーパー大辞林** is built into
> macOS Sierra)_
>

* * *

I work at a university, where it is now winter break. The students at the
university get about 4 weeks off of class between semesters. The staff gets
about a week and half off, during which the university actually shuts down
(conducts no business). How would I represent each of these points-of-view
(students vs. staff)?

  * My gut tells me they would both be 休学 or 休校, but for different reasoning for students vs. staff.
  * Or would the students use 休学・休校 since they are just "resting" from class in 冬休み, but the staff uses 閉学・閉校 since business is ceasing?
  * Or something else...
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synonyms kanji-choice prefixes
Closed:　[休]{きゅう}〜 vs. [閉]{へい}〜
Most commonly used. temporary closure vs completely close  
閉店 It never open again.  
休業 Take a break for a while.May be tomorrow or next month will open again  

休学 is mean of ”Temporary absence from school”. i.e He has been absent from
school for six months because of illness  
休校 is "close (a) school" i.e There are no classes today.  
冬休み winter vacation. it's same mean of english.  
  
閉館 It is necessary to read before and after sentences.  
図書館は5時に閉館します。Library closed at 5 p.m.  
あの図書館は数年前に閉館した。That Library was closed several years ago.
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Your observation is mostly correct, as 休 suggests "off from usual activity"
thus expect returning at some point. 閉 means two things: "closed" as opposed
to "open", and "close down (completely)".

For stores and libraries that have regular business hours, `閉X` could mean
either "end of a day; outside hours" or "close down to cease to exist", but I
believe universities don't have clear open-close cycle, so 閉学 or 閉校 will
exclusively mean the latter.

Moreover, there are special terms:

  * 休学: student takes leave of absence granted by the school
  * 休校: provisional shutdown on special occurrences e.g. disasters (臨時休業)
  * (cf. 停学: suspension from school as a penalty)

As for regular vacations, we only use 休業, 休暇 or 休み that indicate "day off", as
夏季休業, 夏季休暇 or 夏休み. The organization tend to use 休業, and the staff 休暇 or 休み.
There's no distinction between students, faculty and staff as far as I know...
my college use 窓口休止 for being out of administrative service as opposed to
educational semester.
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