No, there isn't really any difference in the virtual photons which are
exchanged between like and unlike charges, but the answer is that
photons are called "vector" particles -- they have spin 1 and their
propagators in relativistic quantum field theory have 4x4 components
(only four of which are not zero, but that's enough). Furthermore, the
interaction vertex of a photon and an electron, for example, has an
associated Feynman rule which is sensitive to each of the components of
the photon field.
In a calculation of the scattering matrix element of two
like-signed particles in relativistic quantum electrodynamics, a minus
sign appears in the matrix element relative to the same calculation
with two oppositely-signed particles, for a single photon exchange.
It's been noted that this isn't enough to generate an observable
difference in the scattering of like- and unlike-sign particles. After
all, quantum mechanical matrix elements are squared in order to compute
probabilities of observing outcomes. The last ingredient is to notice
that the amplitude for the scattering process interferes with the
amplitude for no scattering -- you add the matrix elements for the two
processes and you square the result. No scattering corresponds to a
matrix element of +1, and the sign of the scattering amplitude relative
to that now makes a difference.
Two observations: If the energy of the scattering is high, then
the initial and final states are very different, and the contribution
to the scattered final state from an unscattered initial state becomes
very small. In this limit, attractive and repulsive Coulomb forces give
the same measurable behavior. Rutherford scattering gives the same
differential cross section without regard to whether the force is
attractive or repulsive. Only if the energy of the scattering is low
and you can watch the charged particles move towards or away from each
other can you determine the relative sign.
The second observation is that the interference effect mentioned
above and which is advertised as the big difference between attraction
and repulsion is an incomplete argument without the flipped sign of the
scattering matrix element. The reason for this is that the exchange of
scalar particles (for example, pions, whose exchange holds protons and
neutrons together in a nucleus) is always attractive, even when the
participating particles are exchanged for their antiparticles. Pions
don't have enough components to their fields, and the interaction
vertex of a pion and a fermion like a proton doesn't involve any
components. This interaction is always attractive, even though the
interference argument above holds for it too.
Sources: Peskin and Schroeder, An Introduction to Quantum Field Theory, and Schiff, Quantum Mechanics.
(republished on 07/21/06)