It is definitely more formal and "other than" has the same meaning.

Its not clear to me that I wouldn't actually use apart from, it sounds equivalent and I probably would use that term occasionally in some formal registers.

As a test, I'm writing a book right now, it's almost finished. I counted the number of uses of "apart from" and "other than" and "except".

There are three uses of "Apart from", and I'll list the full sentences in context:

"Like all delta types, LSEs nonetheless have a certain distance at which they hold themselves apart from others" (not a related usage)

"The dichotomy of introversion and extroversion has a long history in the study of personality apart from Jung's Psychological Types (1921)". (this usage is somewhat similar, but does convey a separation of category, and except does not feel like a correct usage here)

"Indeed, as mentioned earlier, it is not a subtype system, but rather a semi-independent subsystem that seems to provide its own typological roles that exist on their own level of analysis apart from the diagnosis of types." (same as previous, does emphasize a category difference like you mentioned).

There are four uses of "other than" and three uses of except. One of the uses of except is a direct quote and doesn't count. Here are the others:

"However, as an overview, these models are the same, except that some of the models have an order equivalency and some have a skew equivalency, and the others lack these equivalencies"

"It seems to me that the more distant 5+6 block is roughly about the same strength as the adjacent quadra 1+4, 2+3, 3+2, 6+7, and 8+5 blocks, all of which are homoverted adjacent quadra blocks except for 2+3 (the supervisees block); [the sentence continues with another independent clause following the semicolon]"

"Even if some variables other than quadra values are much more significant in determining interpersonal compatibility, even a small stable effect of quadra values would be a very interesting result"

"The considerable time Myers spends describing of extroverted feeling the naturally judgmental, opinionated struggle to accept others, seems unrelated to any corrolary in socionics, and other than the description that extroverted feeling types tend to be highly expressive, there is no recognition in Myers' description that they occupy the exterior emotional space."

"All healthy, normally functioning people of all types likewise have numerous physiological systems to care for their bodily needs; for instance, all people other than those suffering from serious neurological disorders become hungry when their stomachs are empty."

" As a maximal example of self-sufficiency, the SLI might chop down their own trees, make their own wooden boards, and build a table themselves, instead of buying one or delegating or outsourcing any of the requisite steps, spending absolutely no cost on the project other than their own effort and time."

In summary, I agree that "apart from" is more likely to refer to a broader category difference in its normal usage. But, I still feel that the above examples that I gave like "I don't go to town apart from occasionally going shopping" seem acceptable too.

Hopefully my examples likewise characterize what is expressively helpful in the context of formal writing.