My Dunmer spellsword (avatar pic) wears light. Most often the Penitus Occulatus fortified with restoration and magicka regen, and Stormcloak officer helm with fortify destruction. Main combat style is destruction at range, unless it's against a mage, when I'll use a ward to get close. CQ combat is dual wield or 'sword and ward'. Follower is Aranea

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Battlemages wear heavy armour.

LA, resto, alt, and one handed.

Basically I think the main point is that the Battlemage is a mage/warrior (in that order) whos main fighting style is destruction supplemented by summons and weapons. The Spellsword uses magic to augment his swordplay and/or unpredictable magic attacks.

Battlemages wore leather armor in Daggerfall, heavy armor in Morrowind, no armor (the class, even though most in-game Battlemages had steel or similar) in Oblivion. The Spellsword used chainmail in Daggerfall, medium armor in Morrowind and heavy armor in Oblivion. Personally I feel that light or medium armor with a shield is more suited for a Spellsword who needs to maintain a good balance between offense and defense.

You could do a mix. For my mage I wear arch-mage robes plus glass gauntlets and boots and I think it looks pretty cool. It's also a good balance of armor and unarmored because i want protection but i don't think a mage should be in full armor

Light armor...never fight in a dress...

Battlemages are supposed to be extremely heavy magic hitters with very strong armor as well. He does not move very well but he has no need of that. Think of a tank or gun turrent. Build-wise, perk destruction, heavy armor, smithing and enchanting. Try to go for 100% destruction cost reduction.

Swordspells are supposed to be light and fast on their feet. So they wear light armor and Instead of shields, they use magic for both defense and offense. Build-wise, perk conjuration, alternation, one-handed, archery and light armor. Alternation to paralyze your enemies and cast flesh-armor spells. Conjuration for bound sword or bow and finishing you opponent off.

No. Spellsword npcs in both Oblivion and Skyrim are skilled in Heavy Armor.

And yet we have a single Spellsword NPC wearing armor (and it's light) while there are 7 armored Battlemages (all in steel and iron gear). Note that this doesn't count any generic NPCs, just named ones.
In Skyrim I know of one Spellsword in light and one in heavy armor (and there are no Battlemages).

Sulla trebatius is in Heavy Armor and uses one handed, destruction, restoration. Kresh the clean and Orchendor are spellswords and although theyre not making use of their skills, they are skilled in Heavy Armor, One Handed, Destruction and Restoration.

Go heavy armor, and put perks on enchanting to best use of Absorb Magicka... absorb magicka weapons on right hand and fireball/heals on left.