Reading around the forum, I noticed that people sometimes compare this Kodak film to CineStill's BwXX and another from Ilford (HP5 plus, I believe).
Much of this discussion AFAIK revolves around the eternal search for a favorable compromise between price and (experiences/subjective) product quality. Keep also in mind that distribution varies across the world, making e.g. Ilford around 30% cheaper here in mainland EU than TriX, while it's pretty much the other way around in the US. Double-X is basically budget proposition and does not necessarily offer anything technically/objectively superior to TriX or HP5+ (let alone TMY2 or Delta 400). Subjective differences can of course play a role, although I've not seen many people express a preference for DoubleX based on product quality arguments alone (i.e. price seems to always be a factor).
Are there common situations in which a 35mm photographer should consider skipping Tri-X in favor of something similar but more suitable for that situation?
If you're happy with what you've got, there's no reason to skip anything, really. I think many people (including myself) like to get a taste of the grass on the other side once in a while to sample its greenness. Personally, I always find confirmation of what I can read in other people's experiences, datasheets etc. - no surprises.
If you're looking for an objective step ahead, a T- or Delta-grained film is arguably better - but that's not to say you will necessarily like it better. Something like TMY2 is noticeably finer-grained than TX and has less of a toe, so it's quite linear even in the deepest shadows. Again, whether such differences are relevant or preferable to you is very subjective.
PS: speaking for myself, my fussing over technical matters and experimentation with materials has always had far less of an influence in my actual photography than things like reflection, practice, discussing images with others etc.