Elemental damage means elemental damage. If you have increased elemental damage then it will increase all the fire, cold and lightning damage you deal, regardless of whether it's with spells or attacks.
Spell damage refers to all damage dealt by spells, regardless of type.
For example, detonate dead is a spell which deals some physical damage - that would be increased by spell damage increases, but not by elemental damage increases.
Likewise, Glacial Hammer is an attack, not a spell, so none of it's damage is increased by spell damage increases, but since some of the damage is dealt as cold, that would be affected by elemental damage increases.
Not all elemental damage is dealt by spells, and not all spell damage is elemental.