While it is true that the ordinary GW invariants of a K3 surface are trivial (by the deformation argument you cite), the reduced GW invariants are non-trivial and capture a lot of interesting enumerative information and structure. The reduced invariants are obtained by modifying the obstruction theory on the space of stable maps and result in curve counting invariants which are invariant under deformations of the K3 surface which preserve the (1,1)-type of the polarization. The generating functions for these GW invariants can be expressed in terms of quasi-modular forms. http://www.ams.org/journals/jams/2000-13-02/S0894-0347-00-00326-X/S0894-0347-00-00326-X.pdf
Donaldson-Thomas invariants are only defined for threefolds. In order to define them for a surface, one must create a "local surface" --- the threefold given by the total space of the canonical bundle over the surface. This introduces some non-compactness which then can be dealt with in various (sometimes equivalent) ways (for example, one can work equivariantly and use localization, or one can use Euler characteristics weighted by the Behrend function in place of virtual classes). After dealing with these issues, one finds that the DT invariants of K3 are indeed non-trivial and are related to the reduced GW invariants by a MNOP type relationship.
This is an extensive subject with lots of work. It began with the formula of Yau-Zaslow and the conjecture of Gottsche back in 1995. Leung and I defined the reduced GW invariants of K3 in our 2000 paper. Recently, there has been a resurgence of work on the invariants of K3 by Pandharipande, Maulik, Thomas, and others, for example: