I would think that a spellsword specializes in dealing physical damage with magical weapons. So conjuring a sword and attacking with it would be more of a spellsword. They specialize in the conjuring skill. A battlemage would specialize in the destruction skill, and deal magical damage, like damage from fireballs and ice bolts. They are the more typical mage-character, and are much more mana-dependent.
A battlemage is a magic oriented fighter, conjuration, destruction, one handed, armor wearing badass. They need a good amount of magicka but also utilize martial weapons and armor