Why do people sometimes put authors with equal contribution in non-alphabetical order?

There are some requirements for funding¹², a degree³, tenure and similar for which literal first authorship counts. If one of the equally contributing authors gains an advantage from being the first author due to this while the other one doesn’t or has a smaller advantage, it can make sense to have the order deviate from the alphabetic one.

Another conceivable scenario would be that the first authors is well-known in the respective field and was made first author to attract a little bit more attention to the paper.

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^{¹ For example several faculties in Germany have schemes for evaluation and publication-based funding that assign special value to first-author publications. Some of those do not mention joined first authorships and do not make sense with multiple first authors, which indicates that first authorship is meant literal (example in German, search for Erstautor).}

^{² This journal, e.g., lists a handful of funding organisations that will pay the publication costs, if the first author is funded by the respective organisation. Joined first authorship is not mentioned. Even if this may be dealt with on a per-case basis, just flipping the first authors may be easier.}

^{³ For example, for a publication-based PhD thesis, it may required that the included publications be first-author publications (example, again in German) without the case of joined first authors being considered. While the latter may be allowed on a per-case basis, flipping the authors may be much easier and avoid a lot of bureaucracy.}

There are a couple reasons for doing this - most of which are rooted in the idea that the "first first" author still has an advantage. Some folks will still refer to the paper by "Adams et al." In fields where first authorship carries weight, some people "don't believe" in co-firsts. There's some ambiguity about whether a co-first author can put their name first on their own CV, which will be important for quick reads by people evaluating said CV.

Given that, there are a couple reasons:

1. "Alphabetical ordering" inherently privileges one of the authors. They may, in the interest of fairness, "flip for it", or in a series of papers just alternate.
2. "First first" might also be the one willing to field some of the post-writing effort behind a paper - actually submitting, fielding press inquiries, etc.
3. There may be someone that benefits more. For example, a postdoc for whom this is a big deal publication may be put first so that they can reap any little residual benefit from those who don't pay attention to the note, instead of an established researcher who doesn't need it as much.
4. Not all journals accept a co-first designation. So the order may be "If it comes down to it, and we can't share, it should be you" decision, even if it does make it into a co-first compatible journal.

One publication I'm on has a author order that involved 2-4.

Two authors who regularly collaborate may have a rotation scheme going (with footnotes about equal contribution). That is, they take it in turns to be first author on any of the papers that is about their joint research. This avoids building up a substantial difference in first authorship, which can be important on CVs where the footnotes are not visible.