It's a weird thing.
While actually using the weapon for damage, it levels up your 1H weapon, and damage is based on your 1H skill. However, casting it and/or having it out around enemies increases your Conjuration level. Perks for both 1H AND Conjuration will work on bound swords, so long as it's either shared accross all conuration/1handed weapons/spells, or its specifically for bound weaponry/the weapon type you summoned.
So, 1H damage increase perks applies to the bound sword, so do the sword perks, etc. And time increases will apply, as well as the special damage increases and anti-summon attributes from conuration.
Basically, all you need to know that Damage is based on your 1H skill, but using it will increase both your 1H and Conjuration levels. And perks from both trees will generally apply.