What are the differences (if any) between the Dyson's series definition and the "in/out" definition of the $S$-matrix

The in and out states are defined as solutions to the Lippmann-Schwinger equation, with the appropriate boundary condition prescribed by the choice of the contour ($\pm \varepsilon$). $$ | \psi^{(\pm)}_a \rangle = | \phi_a \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)}_a \rangle $$ If you calculate the $S_{ab}=\langle \psi_b^-|\psi^+_a\rangle$ you gonna find a recursive relation for the the S-matrix. $$ S_{ab}=\delta(a-b)-2i\pi\delta(E_a-E_b)T_{ab}. $$ $$ T_{ab}=V_{ab}+\int dc\frac{V_{cb}T_{ac}}{E_a-E_c+i\varepsilon} $$ $$ T_{ab} = \langle \phi_b |V|\psi^{(\pm)}_a \rangle $$ $$ V_{ab} = \langle \phi_b |V|\phi_a \rangle $$ If you iterate this recursive relataion over and over you are left with a series. This series can be identified with a time ordering of the desired exponential if you use that: $$ \frac{1}{E_b - E_c + i \epsilon}=-i\int_0^{\infty} dt \exp\left[{i(E_b-E_c)t -\varepsilon t}\right] $$ Then you may see that adiabatic turning on and off interactions are the same thing as putting the $\pm i\varepsilon$ in the Schwinger-Lippmann equation, i.e. imposing the existence of in and out scattering states.

In LSZ approach we go to a different root. We have already a formal hamiltonian (the QFT theory). Scattering states could be build out of this hamiltonian by seeking for long-lived states with some dipersion relation. Mathematically they are poles of the two-point correlation function. The $Z$ is only requiried if you want to work with this fundamental fields inside the hamiltonian, like the fields in the correlation function. Then, you need to assume that the field create and destroy different states than the scattering ones.

The adiabatic assumption is behind all this approaches. Is ultimately related to the fact that exist scattering states behaving as a free particle. The difference is that in Lippmann-Schwinger approach you have asymptotic physical particles and in LSZ you have self-interacting bare particles giving rise to a physical one.

Nogueira's answer was really helpful, I just wanted to add a few remarks in hindsight.

When I wrote this question, one of the things which confused me was that I couldn't see how the "in" and "out" states which one defines in formal scattering theory, via e.g. the Lippmann-Schwinger equation (see Nogueira's answer), would coincide with the states created by the "in" and "out" fields from the vacuum, e.g.: $$\vert \mathbf p \text { in}\rangle=i\intop \text d^3{\mathbf x} f_{\mathbf p }^* (x) \overleftrightarrow {\partial _0} \phi _{\text {in}}(x)\vert 0 \rangle.$$

If one defines the "in" field, for the scalar case, via the linear combination: $$\phi _{\text {in}}(x)=\intop \text d ^3\mathbf p \lbrace a_{\text {in}}(\mathbf p )f_{\mathbf p}(x)+\text {h.c.}\rbrace, $$ and, in turn, defines $a(\mathbf p )$ as the destruction operator for the "in" states, then the correspondence is tautological.

However, some texts (for example, Bjorken&Drell), start from "in" and "out" operators defined by the Yang-Feldman equations:$$\phi _{\text{in}}(x)=\phi (x)-\intop \text d ^4 y \Delta _{R}(x-y)j(y),$$ where $\Delta _R$ is the retarded Green's function for the Klein-Gordon equation and $\phi$ is the (renormalized) interacting field, which satisfies: $$(\square +m^2)\phi =j.$$ In this case, the correspondence is not immediately obvious, since one has to prove that wave-packets constructed from $\phi _{\text {in}}$ converge indeed (at least weakly) to free states.

Precisely this questions are addressed in an old article by Schweber S., "On the Yang-Feldman formalism", and also (quite easier to read) in his book "Introduction to quantum field theory", sec. 17d, where he defines the in field by $$\phi _{\text {in}}(x)=e^{iHt}\Omega^+ \phi (\mathbf x ,0)(\Omega ^+)^\dagger e^{-iHt},$$ and proves that is satisfies the Yang-Feldman equation.