Both use magic. Both use weapons.

The question is, what's the distinction between the two? This is something I've never really understood. Is there even a need for a distinction?

Battlemage-Heavy Armor,One handed(Generally Axes),Alteration and Conjuration.
Spellsword-Light Armor,One Handed(Generally Swords).Destruction and Restoration.

The above.

Though some Battlemages use Two Handed weapons.