Destruction's main issues I believe stem from the prohibitive cost of midgame spells which has a number of effects on the player's progression:
1) Destruction levels up too slowly - The progression of your Destruction skill is slow as you're required to constantly cast destruction skills to level it up. As Destruction's midgame spells are overcosted you are not realistically able to do this. This slows your progression to the next tier of spells as well as slowing your skillups in destruction which reduce the cost of spells in that tree. This leads to a frustrating experience for the player in the mid game who spends most of a fight waiting for mana to regen instead of fighting.
2) Dual casting perk is not worth it early on - The dual casting perk is not mana efficient and further exacerbates the mana inefficiences of the destruction tree dishing out 220% spell damage for 280% spell cost. This perk is actually worse for the mage to invest into until they have enough gear available to reduce the cost of destruction spells to zero/near zero and are able to unlock the Impact perk. This in part is the reason I believe a lot of mages experience so many problems with mana in the midgame. A tooltip that accurately describes what the perk does might help mages to avoid it until later in the game.
3) Enchanting is 100% required to be an effective destruction mage - Due to the overcosted nature of destruction spells the only way to play a destruction mage in a rewarding manner currently is to layer on gear enchanted to reduce the cost of destruction spells to compensate of the cost of the spells. This makes mages a pain to play until you get your enchanting to the point where you can put high level enchants on helm, ring, necklace and armour. Either tweaking the cost of the midgame destruction spells or increasing the availability of gear that reduces the cost of spells (Specifically destruction spells) in the early to midgame would definately deal with this issue.
On the flip side though, once you get past the midgame, have levelled your enchanting and have 100% spell cost reduction for the destruction tree the game becomes incredibly easy. I don't think having all your destruction spells become free in the late game is a good way to deal with the problem and would love an alternative which keeps the game fun. Possibly making the effect of the spell cost reduction enchants multiplicative instead of additive and tweaking the cost of all destruction spell costs would leave the late game mage fun to play while balanced.
TL;DR - Destruction spells cost too much so mage play style revolves around leveling enchanting to the point of making the spells free.
EDIT:
Had a few people asking about what gear to use to reduce the spell cost of destruction spells to zero. Full details http://www.gamesas.com/index.php?/topic/1283388-destruction-is-not-underpowered-destruction-spells-are-overcosted/page__view__findpost__p__19376023.