The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and *that's* what the "n-dimensional field theory" plays with (for each n). For GW we're analyzing the marked/nodal points on surfaces and their moduli, and *that's* the playground of the 2-dimensional field theory (independent of n). **So in each case (SFT versus GW) the field theory structure is about different things.** The algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, *"Symplectic field theory and its applications"*.

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-cobordisms. It recovers Taubes' Gromov invariant, and there is a forthcoming paper of Hutchings on "ECH as a field theory" (you can find a sketch now in his blog posts). This is expected, because ECH was inspired by SFT and Taubes' Gromov invariant.