I am a quite confused on the differences between these two formulations of open string field theory, and am unable to reconcile how they are related. M. Kaku's 1972 paper formulates a string field theory based on second quantization and calculates a propagator and scattering amplitude with this. However, Witten's cubic open string field theory seems, at first glance, quite different. Is there a way to connect these? Is the answer just to say, "Well, the scattering amplitudes are the same for both of these separate pictures, so they are the same"?
The relation between covariant and lightcone gauge string field theories is unknown.
In principle, the relation should be analogous to the relation between the path integral and lightcone gauge Hamiltonian formulations of Yang-Mills theory.
For free string field theories, this analogy can be made precise. At the interacting level, things do not work so easily. The main difficulty is that covariant string field theories contain an infinite number of time derivatives in their interactions, so one does not know how to pass to the Hamiltonian formalism and impose canonical commutation relations. A related problem is that the interaction vertices in covariant and lightcone theories have a very different structure.
My guess is that the infinite time derivatives in covariant string field theories is an artifact of the choice of field variable. With a field redefinition it should be possible to obtain a gauge invariant (but not covariant) string field theory which is local in lightcone time. Then one should be able to fix lightcone gauge and pass to the Hamiltonian formalism to obtain the lightcone string field theory.
I only have the haziest idea of how one might go about constructing such a field redefinition. This is not something people think about these days, and generally there are very few people working on string field theory. However, I think this problem is a "slumbering giant" whose solution could have a big impact.