Why should I use a bound sword?

I am playing my first character in Skyrim and I decided that I want to focus on conjuration. Up to now I used swords for close combat situations and transitioning into a bound sword seems to be a good choice. I looked up the damage a bound sword can inflict and it seems like that even the mythic version cannot do more than 14 damage per hit. My current sword already has a damage value of 35, more than double of the bound mystic sword. In addition I lose the possibility to enchant the sword or improve it through smithing.

On the upside I can use the perk to automatically cast soul trap which I really like. I can also use the bound sword to improve my conjuration skill. I don't use any skill boost tricks, so I need to level up conjuration naturally and the bound sword comes in handy for that.

Are those upsides enough to invest into the use of bound weapons? The lack of damage seems to be a real issue.

There are a few reasons why you would want to use one:

1. It saves you weight from not having to carry one around.
2. Like you pointed out, with perks, it can automatically soul trap an enemy, so you can use a better weapon unless you need soul trapping.
3. With even more perks, you can banish summoned enemies. Note both of these can be reproduced via enchantments, but this way you don't have to recharge your weapon.

The drawbacks:

1. Yes the base damage is high, but you cannot improve a bound sword via smithing, so the damage is really quite low during the end game.
2. It lasts only so long, then you'll have to recast.

All in all, like you said, the lack of damage does become a big issue once you start investing into smithing.

The 14 is base dmg which means that it changes as you level up one handed! It has 14 dmg if you are lvl one in one-handed.
14 is A LOT of base dmg; in fact, it's the second best sword in the game as a daedric sword has 15 base dmg. So basically it's a good sword but at higher levels you may want to start using weapons and magic like a battlemage. Please remember bound weapons do Less damage to dragons because dragons have magic resistance.

For mages, there really isn't a point except to farm souls for soulstones. As a mage you want to keep your distance from your enemies. Plus, as a mage, your skill in one-handed weapons won't get as high to take advantage of the the skill bonuses. If you have a big ugly in your face, I think you're best off with some sort of enchanted weapon instead.

These swords are pretty good with all the perks and the One-Handed and the backstab perk. It comes with the enchantment of soul trap and banish so its good to use if you're need souls. Main problem is that it lasts 60 seconds and costs magicka. If you use one have the Unrelenting Force shout ready

I love the bound sword. It reduces weight and fills up soul gems rather quickly. In fact I use both the bound sword and bound bow. They are very useful when fighting necromancers and conjurers if you have the oblivion binding perk as it banishes atronachs and turns raised undead, though this does not work on unbound undead only ones raised by a spell or the ritual stone. Add to the fact that you can get it near the very beginning of the game and it makes for an excellent weapon.

There are 2 issues with it:

First, if you decide to smith, which I do, you cannot improve the blade. This causes it to be left behind by other blades if you improve them.

Second, you cannot enchant this blade. Enchanting a blade allows you to customize. For example when I get to daedric smithing I smith a daedric sword improve it then dual enchant it (you must have the perk) with fire and storm. This causes an extra 64 points of damage if you have all related perks and use a grand soul gem. It also deals 32 points of magicka damage making it extremely powerful.

All in all the bound blade is a good sword to use until you get daedric or dragonbone unless you want a custom enchanted blade. Also if you hesitate to switch due to losing soul trap enchantment enchant a weapon with soul trap and give it as well as empty soul gems to your follower. If they use the weapon they will fill the soul gems.