Elven powers pretty much grow over time (is there any downside to being an elf?). Arwen is less powerful than her father, Elrond. But even Elrond is no match for Galadriel, who was born before the First Age of Middle-earth. She could be close to 10,000 years old.
Unlike Arwen, Galadriel's age isn't portrayed by a deepening of her voice but by her possession of knowledge and sight, which seems to penetrate both the minds of the fellowship and the whole future of Middle-earth. Galadriel is the bearer of Nenya (one of the elven rings of power) and therefore has a connection with Gandalf and Elrond, the other elven ring-bearers.
Galadriel is at once gentle and frightening. We see her looking gently upon each member of the fellowship as they prepare to embark on their journey. She smiles to them and clothes them in elven cloaks and brooches and gives them each a special gift. But Galadriel also says some terrifying things about the world being on the edge of a knife… not to mention her late-night rendezvous with Frodo, when she grows dark with her desire for the power of the Ring and her voice warps into something almost demonic.
But Galadriel passes the test that she must have seen coming. She denies herself the Ring when Frodo offers it, and says that she will leave to the west. With even the ancient and mighty Galadriel departing from Middle-earth, the time of the elves is truly at a close.