What (if anything) is the difference between bullsh— and horsesh—?

Late answer on this one, but less formal resources than the print dictionary seem to agree with my impression that "horse--" is more emotionally charged and focuses more on a situation that is abusive or unjust.

Urbandictionary sense 2 has:

as opposed to bullshit, which expresses judgement on shallow, insensible, and false ideas, horseshit puts more emphasis on the speaker's distress about the message just received

Wiktionary sense 1 has:

Serious harassment or abuse

(As a methodological side note, I'd argue that when looking for the definitions of expletives, community sources or less-official sources often have more current and nuanced information than publication-ready dictionaries.)

Essentially, there are two meanings with which bull-- or horse-- are used. For bull--, the more prominent one is "something said without regard for whether it is true or false, in which the speaker has no sincere belief." However, a secondary meaning is "unfair treatment or an unjust situation" (which you'll see contextually from a google search for sports). This meaning is especially invoked when used with an intensifier, "f--g bull--".

My impression--and others would seem to agree, see a blog posting about the distinction in which a commenter succinctly argues "Bull[] is something that is not true and horse[] is a situation that is unfair"--that for horse--, the secondary and primary meanings are reversed. With horse--, the "unfairness" meaning is prioritized, even though it can also be used for bull--'s primary meaning of statements said for convenience rather than truth.

Here's another contextual usage example: this is a T-shirt which has the choice pull quote, "F--g horse--", from the CEO of the Dallas Stars, a hockey team; contextually, he was reacting to the 2018-2019 season, during which two very highly compensated players were under-performing. Horse-- conveys a greater sense of disgust, and can do so just fine on its own; see its use in Titanic, in a scene in which Leonardo DiCaprio's character Jack is reacting with anger and disbelief to being framed for grand larceny--the actor's line delivery is typical of the greater emotional charge.

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Similarly, as of the morning of June 10 2021, here are the top 10 hits for a Google News search for the term "bull--":

1. Why the bull--jobs thesis may be, well, bull--
2. 'This Is Some Bull--': Women's Group Rails Against Biden Inaction on Equal Rights Amendment
3. Nation Rocked By 40 Million Acts of Total Bull-- (The Onion)
4. "They talk about national sentiment, but that's bull--"
5. Book Review: 'Calling Bull--' will help you be a better advocate
6. Fox Business Host Rolls Over When Trump Spews Election Bull--
7. Katt Williams Thoughtfully Explains Why The Cancel Culture Narrative is Bull--
8. The bull- jobs theory may turn out to be, well, bull- (second source reporting on the same study)
9. How to Tell If Your Company's Pride Campaign Is Bull--
10. Brooks Koepka lost his train of thought hearing [someone's] bull--

Of these, headlines 2 and 3 are used in the sense of "unfairness," while the rest are used in the sense of "false or nonsense statements."

By comparison, the top 10 hits in Google News for 'horse--' are:

1. QAnon's cowboy-themed "Patriot Roundup" sure sounds like it was full of horse--
2. Farmers Only: Dayton comes back from huge deficit. [from blurb, "Brantley 'Ding Dong' Bell singled, walked twice, and scored three runs while Aristides 'Horse--' Aquino has turned into a walk machine"]
3. After Wednesday's blowout, the Orioles June magic is on hold, for now [from blurb, "Harvey himself said to reporters, 'It's been a couple years now of being really horse--'."]
4. The horse-- paradox: Why fears about tech are wildly exaggerated
5. Your Apple AirTag can't say horse-- but it can say poop horse
6. Samantha Bee Says We Need To Strangle The Filibuster to Death [From blurb, "It's horse--", referring to the filibuster being unjust]
7. Prison Break Season 6 canceled but why are fans applauding the decision? [Blurb: "Another user comment: 'Season 5 was complete horse--, so I'm glad there's no season 6 to be honest'"]
8. Thoughts on a 4-2 Rangers loss [Blurb: "Gallo having homered earlier in the inning, was a horse-- call... I mean, that's horse--"]
9. Cut after 31 minutes: Heal on the outer after shocking move [Blurb: "then cutting a 19yo for lack of experience is horse-- as a GM, as a coach, as an organization and even more so, as human beings"]
10. In Leaked Video, Mayor Garcetti Compares Unhoused People to Horse--

Of these, only hit 1 refers to "something false or nonsensical". Hits 2 and 7 probably mean "something worthless" (one of them being used as a nickname--I don't know if this was the writer being cute or if it's actually used, baseball is weird). Hits 4 and 10 refer to literal horse excrement, hit 5 referred to the word "horse--", and hits 3, 6, 8, and 9 are referring to an unfair or unjust situation.