What's the difference between "based on computation of the odds" and "based on some model"?

Taleb is doing some handwaving here.

"Some model" in this context is just the assumption of a specific probability distribution. So if, for example, you believe that the observation values are normally distributed with the mean of 0 and the standard deviation of 1, the chance of seeing a value greater than 3 (a "three-sigma value") is 0.13%. The chance of seeing a value greater than 6 (a "six-sigma value") is 9.87e-10. E.g. if your observations are financial daily returns, you effectively should never ever see a six-sigma value. The issue is that in practice you do see such values, pretty often, too.

The problem with Taleb's statement is that to estimate the probabilities of seeing certain values in the future necessarily requires some model, even if implicit. Without one you can not do the "computation of the odds" unless you are happy with the conclusion that the probability to see a value you've never seen before is zero.

Taleb's criticism of the default assumption of normality in much of financial analysis is well-founded. But when he starts to rail against models and assumptions in general, he's being silly.