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Consider the following sentences:

  1. Try not to be alarmed if a rule doesn’t seem to work for a specific sentence.
  2. Try not to be alarmed if a rule seems not to work for a specific sentence.

Is there a difference in meaning between "does not seem to work" and "seems not to work"?

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No, there is no difference.

Seem is a verb that governs infinitive complements and allows Negative-Raising. That means that negation in the infinitive complement of seem, or want, or other Neg-Raising verbs, as in

  • The rule seems not to work.           [ = ... to not work]
  • He wants me not to go tomorrow. [ = ... to not go tomorrow.]

can also appear, instead, in the matrix clause with seem or want

  • The rule doesn't seem to work.
  • He doesn't want me to go tomorrow.

without a change in meaning.

This is not true of most predicates, which don't allow Neg-Raising. (Be) Easy, for instance, is a more normal predicate; the two sentences below do not mean the same thing.

  • It's easy for him not to smile.
  • It's not easy for him to smile.

Edit: Pursuant to RegDwight's comment above on split infinitives, I should mention that both the unsplit variant not to smile and the split variant to not smile are in the complement clause, i.e, not Neg-Raised -- though they are likewise equivalent in grammaticality and meaning. Either form can be regarded as the "source" of the Neg-Raised not easy to smile

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  • I suspect it's not a common practice, but I almost always use these two phrases to distinguish between different knowledge claims, and it comes up in my writing all the time. If something "does not seem to X," it fails to present evidence of X. And if a thing "seems not to X," it presents evidence that it doesn't X. That is, the latter indicates presence of confirmation, and the former indicates absence of disconfirmation. In both cases, "seems" signals that the observation is not sufficient to settle the matter of whether thing does X.
    – Tom
    Commented Apr 9, 2021 at 7:10
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    @Tom Such distinctions can be made, and if one is careful in making them, one can train one's readers to recognize them. But the contrast is very pointillistic, and most people wouldn't notice it unless you styled the text around it, as you describe.
    – John Lawler
    Commented Apr 9, 2021 at 15:00
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    Are you saying I can't take my pet distinction "on holiday"? ;)
    – Tom
    Commented Apr 9, 2021 at 15:08
  • Check and see first what the local leash laws are.
    – John Lawler
    Commented Apr 9, 2021 at 15:22
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There is a slight difference.

Clearly both sentence do not say that "the rule doesn't work". They talk about a preliminary examination and your conclusion so far, which may turn out incorrect in the future.

"It doesn't seem to work" is a slightly weaker statement. The speaker hasn't put much effort into the examination so far. He or she hasn't seen evidence that it works yet, but hasn't looked much at the problem. "It seems not to work" is a strong statement. The speaker believes that it doesn't work, but accepts the possibility to be wrong. That said, there isn't much difference.

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