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Many go on to use their fame to develop various scientific directions of research, and it is no surprise that careers mostly go downhill after a Nobel: you can only top the mountain so many times, and even if what you do later is fine, it won't be at the same level. And, of course, besides hearing all the flattery and being offered all the resources, positions, labs, and distractions, you also get older. Real innovation is largely the realm of the young. Some age gracefully, or use their influence for good causes though perhaps unrelated to their prizewinning innovation (Einstein and Pauling come to mind). But not all.
Poor Watson and Crick!
Francis Crick went on to speculate about extra-terrestrial origins of life from spaceships and to somewhat over-extend about how well we might understand consciousness by experiment (on monkeys). Watson went on to a career of solid success, if not transformative discovery, in molecular biology and science administration. But in many ways, over many years, he showed that no matter how impressive his understanding of molecular biology and its complexity, his elementary simplicity about other subjects was comparably spectacular, but because of his fame, not much inhibited. As a result, he has fallen from grace in a way that he has earned.
Like too many hard-core scientists, geneticists and physicists alike, he felt he also knew everything there was to know about humans and, even more so, about human society. But, in voicing his opinions about racial differences in intelligence, he made a very common, if wholly culpable, mistake. This may have been intentional but it's far more likely that it was just typical, ignorant, racist folklorish thinking, coupled with a lack of self-restraint.
Thus, according to Watson, whites are smarter than blacks. What he seems to have meant based on his phrasing, and what most people have in mind when they say this sort of thing, is represented schematically by the left part of this figure: two non-overlapping distributions. This represents a common problem when two groups are compared.
Sometimes the two groups are cases vs controls, or 'overweight' vs normal weight. It could be the effects of some environmental exposure--calories, gluten, cigarettes. Or it could be a behavior, with sexes compared, say. So, with respect to the above figure, one distribution would represent, say, smokers, or people who are obese, or people who drink alcohol, or whatever, and the other not. But often the categories are 'races', and as in Watson's case, they are the usual categories ('black' vs 'white', Jew vs non-Jew, Asians vs Europeans, etc.), and the trait is a common one (e.g., intelligence).
The image such categorical statements conjure up is, as on the left of the figure, that every blue is 'better' (meaning, their score on an increasing axis as shown) than all reds. Thus, for those who declare that "blacks are not as intelligent as whites, and anyone who has employed blacks knows what I'm talking about" [a near-quote that has been attributed to Watson], that is what you are implying even if not saying that in so many words. IQ tests are also commonly cited in this way and in this regard. Never mind what 'black' and 'white' mean in our complex, admixed, highly socioeconomically partitioned world. Never mind that as testing improves to be less culture-bound, the gap between the mean scores of these already vague and artificial categories has narrowed ("Black Americans Reduce the Racial IQ Gap", Dickens and Flynn, 2006; Psychological Science, vol 17, no 10, p 913-920) and that many testers do not hold to the reality of such mean differences, or the concept itself as a biological real entity, but rather view the differences as artifacts of test construction.
Here, let's even grant, for argument's sake, that there is a true, inherent, even genomic mechanism that wholly determines intelligence or your other 'favorite' comparison trait. Even then, we never see absolute non-overlapping distributions. What we see is like the image on the right side of the figure above. Group mean differences, but huge overlap (the image is of a 1 standard deviation difference in the means, which is greater than most modern IQ-race comparisons). Seeing such a graph one would not, or at least responsibly should not, make the categorical-difference assertion, that "blues are more intelligent than reds". It simply is not true that even low-scoring blues are higher-scoring than every or even most red. Yet that is the impression that is given, and it's not excusable even as an an inadvertent shorthand.
Traits have distributions and they usually overlap. It is fair to expect that someone like Watson should know better than to think otherwise. And, as a person of exalted reputation, and presumably knowledgeable about things genetic, he should absolutely refrain from making stupid categorical statements even if he is naive enough to believe them, or believe they are harmless over-generalizations. His reputation places a burden of responsibility that most of us don't have to bear (but then, we didn't get Nobel Prizes, either).
When a James Watson says the kinds of things he has repeatedly said, his statements can have consequences. Saying, e.g., that western policy (e.g., foreign aid) to Africans is misguided because they can't live up to our standard, whatever that means, may well influence policy, given the kinds of politicians now in the US Congress, with potentially lethal implications for countless people. It's a serious mis-use of science to make categorical statements when they convey misleading characterizations.
Watson isn't the only public figure misusing and misrepresenting science. But he has a long history of proclaiming his similarly irresponsible views (e.g., this), and he should know better. People making the kinds of statements he makes may protest their innocence, claiming that they are not sexist or racist, but just telling the truth. Perhaps they could just be more open about their personal views (such as who they'd hire), though we do have laws against discrimination in hiring. You have a right to be a racist, but when you are someone like him (who may in fact not be a personal racist), you allow science to be turned into folklore and in turn into fodder for demagogues (such as politicians, one might suggest).
That's a different subject from the current one, which is that these statements inadvertently, implicitly if not explicitly, indulge in misrepresenting traits--thinking of the categorical left-side difference in the figure, even if knowing it's far more like the right side. We see such exaggerations all the time, in terms of genes 'for' disease and group or race categories. Watson is the standard bearer for this sort of thing because of his fame. Many others follow in the parade of simplistic misinterpretation, for reasons that are only sometimes innocent misunderstanding of the data.
Not so elementary, my Dear Watson
After many statements of these sorts, Watson simply (or, one might say, elementarily**) earned a lot of the censure he's experienced that has cost him some aspects of his job at Cold Spring Harbor. So he claims his income was cut, and he wants to sell his Nobel Prize medal. If his financial straits are as dire as he suggests (so dire, in fact, that he needs to sell his medal to be able to buy that David Hockney painting he's been coveting), then it is because his dumb statements have had consequences, and rightly so.
You might say this is just a matter of political correctness, but that is likely because you're not on the receiving end of the denigration. Mistaking the right figure for the left is not about sound scientific discussion of what the facts are and how to interpret them, areas which themselves are far from viewed with consensus. Instead, Dr Watson's fall from grace is pathetic in the proper sense of the word. By not recognizing the important responsibility of someone who has been given high honors, his current state was well earned. And the inability to buy an expensive original painting is a far cry from the hurt he implicitly wished, innocent though he may feel about it, on a great many people.
*Except that having it made apparently doesn't include the ability to buy expensive original art.
**Though Holmes never actually said this to the other Dr Watson.