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Generally, the EM wave is composed of the magnetic field and electric field.

What is the difference between an electric field dominated wave and a magnetic field dominated wave?

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In the far field (>1wavelength) there is no difference,
because an electromagnetic wave always contains E and H,
both in phase and of equal energy.
Therefore it is called electromagnetic.
In the near field the source dominates the field, e.g. dipole and loop.

That is the reason, why electrical or magnetical antennas do not exist.
An antenna does not draw energy from one of the components,
but from the Poynting vector, which is the product of E and H.