Is "Lórien" just a short form of "Lothlórien" or does each one apply to something unique?
It's the same thing. Lórien is the shortened Lothlórien.
Actually, it was first known as Lindórinand.
Tolkien had given many names for the specific location, each with a different meaning:
Name Meaning Origin Lindórinand Valley of the Land of the Singers Older Nandorin name of the area Lórinand Valley of Gold Nandorin name after the introduction of mellyrn trees Laurelindórenan Valley of Singing Gold Sindarin name after the introduction of mellyrn trees Lothlórien The Dreamflower Sindarin name in the Third Age Lórien Dream Land Shortened form of Lothlórien matching the name of the Gardens of Lórien in Aman
The form Lórinand was also rendered in Quenya as Laurenandë and in Sindarin as Glornan or Nan Laur, all of the same meaning. Other, later names given to the land included the much later Rohirric name Dwimordene (from dwimor "phantom", an allusion to the perceived magic of the Elves), and the Westron name The Golden Wood.
--(Unfinished Tales 1980, History of Galadriel and Celeborn, note 5)
According to Treebeard,
‘Hmm, did he now?’ rumbled Treebeard. ‘And I might have said much the same, if you had been going the other way. Do not risk getting entangled in the woods of Laurelindórenan! That is what the Elves used to call it, but now they make the name shorter: Lothlórien they call it. Perhaps they are right: maybe it is fading, not growing. Land of the Valley of Singing Gold, that was it, once upon a time. Now it is the Dreamflower...
"Lórien" is just an even shorter version.
I would also like to add that in the Tolkien mythos, beside Lórien being a short version of Lothlórien there are also the Gardens of Lórien, the land where the Vala Irmo dwelt. Irmo is usually referred to as Lórien, but all seems to indicate that he received the name from the land, not the reverse.
Since I don't have at hand an English version of the Valaquenta I leave you a snippet from Wikipedia
Originally named Irmo [ˈirmo], but referred to more commonly as Lórien [ˈloːrien], after his dwelling place. Lórien and Mandos are the Fëanturi, masters of spirits. Lórien, the younger, is the master of visions and dreams. His gardens in the land of the Valar, where he dwells with his spouse Estë, are the fairest place in the world and are filled with many spirits. [...] Olórin, or Gandalf, prior to his assignment by Manwë to a role as one of the Istari, was a Maia long taught in the gardens of Lórien.
Lórien is the garden where Irmo, brother of Namo, dwells. He is often referred to by the name of this place; the same thing happens to his brother, Námo, which most know as Mandos. (Silmarillion, pg. 19) It was likely re-named this after the Irmo's dwelling place, which was a garden. It was also a place of healing (Míriel was taken there to rest and heal from her weariness, before her spirit left her body).
Lothlórien does mean Dreamflower, as Treebeard said. Loth is prefixed to it, and Lothlórien means "Lórien of the Blossom." (Silmarillion, pages 408 and 409, in the index)
As Melian spent much time there tending to the trees, and Galadriel spent time with Melian (who protected Doriath in a similar fashion to how Galadriel would later protect Lothlórien), I think she found it fitting to name her realm in the hope that it would be like Lórien. (Silm 147-149)
However, once, it was commonly called Laurelindórenan, and that name is the one Faramir uses. (652)