Question: How does a good statistician contrast with a good mathematician? How do the fundamental skills of the two differ?
My answer: “Statistician” covers two rather different groups of people (although some are in both groups). A theoretical statistician has very similar skills to a pure mathematician. Their job is to come up with and prove interesting theorems. Sometimes, these have real world applications, sometimes not. I’d guess that the proportion is higher for statistics than for math.
But statistician can also mean data analyst and the skills here are very different. I’ve been a statistician for 25 years and I’ve only come up with one theorem (which someone else proved). You can be a good data analyst without knowing huge amounts of math (some calculus, but not much beyond that). But you have to be able to figure out what the problem is, then figure out what tools might help with the problem. The toolbox is very different.