What are the practical differences between correlators and scattering amplitudes in quantum field theory?

On a very practical level: scattering amplitudes describe the evolution of an IN state into an OUT state; how the dynamics is encoded in correlators instead?

I think that one difference in particular is that in a conformal field theory (and therefore also at a fixed RG point of a QFT) scattering amplitudes, introducing external momenta (i.e. scales), seem to break the conformal symmetry, while correlators don't. Is this a problem for scattering amplitudes? Should this be a reason to prefer correlators in a CFT regime?

On a somewhat more technical level:

there are many advanced mathematical results for scattering amplitudes such as recursive equations and algebraic relations coming from polylogs or the general structure of the feynman diagrams, or even more strongly the S-matrix bootstrap (amplitudes are completely determined once asked analycity, Poincarè invariance, locality and unitarity); can these be translated to correlators or are these objects in some way less manageable? If correlator-oriented versions of these properties are possible, how comes that in mathematical physics there is much more focus on scattering amplitudes? (maybe it's just my impression?)

Don't worry if you cannot address this last point in your answer, maybe it would require too much effort.

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    $\begingroup$ I'd say that the correlators are the more basic objects as the particle spectrum and scattering marix can be extracted from them via the LSZ reduction. $\endgroup$ – mike stone Jun 4 at 12:49

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