From an in-universe persective, it's basically
Battlemage: Any mage who uses magic for martial purposes
Spellsword: Specifically a fighter who is both a swordsman and a mage
All Spellswords are Battlemages, but not all Battlemages are Spellswords.
Of course, we also have class descriptions from Morrowind.
Battlemages are wizard-warriors, trained in both lethal spellcasting and heavily armored combat. They sacrifice mobility and versatility for the ability to supplement melee and ranged attacks with elemental damage and summoned creatures
Spellswords are spellcasting specialists trained to support Imperial troops in skirmish and in battle. Veteran spellswords are prized as mercenaries, and well-suited for careers as adventurers and soldiers-of-fortune.
Of course, NPC dialogue is more descriptive, but I don't have the time or patience to find it and write it down.