THE RASENGAN
-The Rasengan does not require any hand seals to use, relying only on the chakra control of the user. Once it is complete, it becomes self-sustaining, which does not give the Rasengan a definite limit like the Chidori. The compacted nature and moving speed of the chakra allows the Rasengan to grind into whatever it comes in contact with to inflict major damage. This tends to launch the target spiralling backward after being hit or, if the target is lying on the ground or being launched to the ground, the destructive force is enough to form a crater. In the anime, Naruto was able to throw the Rasengan, making it a mid-range projectile technique. It has been noted a few times that the Rasengan is superior to its counterpart, the Chidori, by Jiraiya, Kakashi Hatake, and even Sasuke Uchiha after seeing the after effects of the Rasengan on a water tower.
The Rasengan was designed to use the caster's elemental affinities. Minato was unable to add his affinities to his Rasengan before he died, while Kakashi Hatake tried to add his lightning element to his Rasengan, but was unable to and explains that he created the Chidori and Lightning Cutter instead, which require a certain point of shape and nature manipulation itself. Naruto was the first to add his wind affinity to his Rasengan to create the Wind Release: Rasengan, and even advanced it up to the point that he could create the far more powerful and destructive version, Wind Release: Rasenshuriken.
THE CHIDORI
-This technique has a major drawback, however. The speed at which the attack must be done, combined with the fact that the user must run in a straight line, causes a tunnel vision-like effect for the user. This allows the enemy to easily counter the attack, making it a potentially lethal move for the user. Though this drawback would seem to make learning the technique fairly pointless, Sasuke and Kakashi are able to avoid the drawback because of their Sharingan, which takes in every detail, regardless of how fast the user is moving.
As an additional drawback, the large amount of chakra required limits the number of times this technique can be performed per day. In Part I, Sasuke can only use it twice a day (under his own power) and Kakashi can use it four times, though their limits have been shown to have increased to an unspecified level in Part II. Should the user attempt to go beyond their limit, all of their chakra will be drained and their very life force could go along with it.
An additional problem, though not necessarily a debilitating one, is the fact that the large amount of chakra can also injure the user's hand if they put too much force into the attack. In Part I, Sasuke does this when confronting his brother Itachi, causing some of the skin to peel from his hand and leaving it slightly smouldering after the attack was finished.
My conclusion is that the Rasengan is the overall better Jutsu. My reason is because of how versatile it is. Learning the Chidori is pointless unless you poses a sharingan. Anyone can preform the Rasegan with the sufficent amount of training (Konohamaru).