Is there any specific reason for the name change? According to me, Cosmic Cube seems a nice and cooler name.
Because they aren't specifically the same thing. While they share a similar abilities (the power to alter the environment around them, and the generation of colossal amounts of energy) the Tesseract in the Marvel Cinematic Universe is essentially a doorway through space (and possibly time), the Cosmic Cube is an object of immense power that has several vaguely defined abilities. They both can be associated with the mad Titan Thanos and the Red Skull in both universes. The fact that Marvel Studios chose not to explicitly refer to the object depicted in Thor, Captain America: The First Avenger, and Avengers as "The Cosmic Cube" allows for them to use that name to describe another cube if they chose to show A.I.M. creating one in a later movie.
In Thor: The Dark World, it's revealed that the Tesseract is in fact one of the Infinity Gems. Something they were able to do because they previously avoided calling it a Cosmic Cube.
In-multiverse explanation: There are several different "universes" within the Marvel Multiverse, and each of these vary slightly. In some, the X-Men were wiped out by sentinels, etc.
The movies are clearly from a different universe than the main comic book universe. You'd expect differences like this.
The fact that this is an alternate universe is even covered by the Wikipedia article on the Chitauri:
The Chitauri are a fictional race of extraterrestrial shapeshifters that appear in publications by Marvel Comics's alternative universe, Ultimate Marvel, in the limited series The Ultimates. They were created by Mark Millar and Bryan Hitch. They were created for the Ultimate universe franchise in place of the existing Marvel Comics alien species, the Skrulls, which play a similar role in the franchise's mainstream continuity. Later, Marvel would choose to distinguish between the Skrulls and Chitauri of the Ultimate universe. The species would later be adapted to other media, notably appearing in the 2012 film The Avengers as an invasion force led by Loki.
All of that merely shows that there are several different universes in the Marvel Multiverse. In the movie, we were simply in a different reality (namely earth-199999). There's no real reason to even expect that the cube is a Cosmic Cube.
The properties are different, and even the Marvel Comics Database indicates that this is different from a Cosmic Cube from the main continuity on the Cosmic Cube page.
Marvel Cinematic Universe (Earth-199999)
Cosmic Cube (Earth-199999)
ADourAdded by ADour
The Cosmic Cube or Tesseract, as it is mostly called, of this reality is a powerful Asgardian artifact created by Odin which possesses the abilities to create infinite energy, which can be used to create wormholes to other parts of the universe and different dimensions, rather than reality manipulation. Unknowngly why, the Tesseract was hidden on Earth.
Marvel Cinematic Universe is different from the existing comic Marvel-verse. So the Tesseract is an Infinity Gem that looks inspired from the cosmic cube. Even Thanos' end credit scene makes more sense now because his comic counterpart wants to have all six infinity gems for a big purpose.
If, in fact, the Tesseract is the cosmic cube appearing in Marvel comics, then the Thanos character would only need to acquire this artifact from Loki and his desires would likely be fulfilled. He would not need to send the "other" to enlist Loki's aid to conquer Earth, the cube would accomplish this task quite easily. This cube, the Tesseract, is very powerful as a source of unlimited energy, but it is not useful as a complete weapon in itself.