Have you seen this? -This is from the most authentic source 'Naruto Wikia'-
"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."
If you search for chidori and raikiri in the naruto wikia, you will observe that raikiri is derived from chidori. It's just a much more powerful version of chidori, requiring much better shape manipulation of lightning release. Kakashi is so much apt with chidori that he doesn't need to run and form hand seals to perform it anymore. Due to years of practice, he can do this at such a level that he can cut lightning with it, or throw it (using it like a kunai) without loss of speed or momentum, unlike chidori, and so he renamed it as raikiri. Its just that Raikiri is much more powerful than lightning unlike chidori, which helps it to cut through the insulation of air effortlessly and efficiently. Sasuke can't perform it at that level, and so his attack is named as chidori. The difference in power is clearly indicated from the white tinge of chidori and blue tinge of Raikiri. Cleared the doubts yet? Still, although the raikiri increases long range versatility a lot (like Rasen shuriken counterpart of Rasengan, with less chakra but more concentrated), the after effects of this attack is still similar to that of Chidori, its just that it increases the depth of cut to a great extent. So, for a wide area damage, its still a win win for Rasengan nonetheless.
