I don't know if this is what the poster is thinking of, but one example that came up recently for me is the distinction between risk-aversion and uncertainty-aversion (these may not be the correct terms).

Risk aversion is the what causes me to strongly not want to bet $1000 on a coin flip, even though the expectancy of is zero. I would characterise risk-aversion as an arational preference rather than an irrational bias, primarily becase it arises naturally from having a utility function that is non-linear in wealth ($100 is worth a lot if you're begging on the streets, not so much if you're a billionaire).

However, something like the Allais paradox can be mathematically proven to not arise from any utility function, however non-linear, and therefore is not explainable by risk aversion. Uncertainty aversion is roughly speaking my name for whatever-it-is-that-causes-people-to-choose-irrationally-on-Allais. It seems to work be causing people to strongly prefer certain gains to high probability gains, and much more weakly prefer high-probability gains to low-probability gains.

For the past few weeks I have been in an environment where casual betting for moderate sized amounts ($1-2 on the low end, $100 on the high end) is common, and disentangling risk-aversion from uncertainty aversion in my decision process has been a constant difficulty.