Legolas' Mirkwood bow was relatively short, and not especially powerful or long-ranged. From Fellowship:
Arrows fell among them. One struck Frodo and sprang back. Another pierced Gandalf's hat and stuck there like a black feather
Legolas turned and set an arrow to the string, though it was a long shot for his small bow.
We can see here that orc bows were powerful and accurate enough to reach the Fellowship at ranges where Legolas' bow would apparently struggle. Later, we get a description of the bow Galadriel gives him:
To Legolas, she gave a bow such as the Galadhrim used, longer and stouter than the bows of Mirkwood, and strung with a string of elf-hair. With it went a quiver of arrows.
A longer, stouter bow is almost by definition going to be more powerful than the short-ranged affair he had been using up to that point. It's not explicit that there was some form of 'magic' in the bow, but given what we know about other elven artefacts it seems likely; in which case, what with the Galadhrim being closer to the elven power of old, one can imagine it has greater virtues than the relatively mundane weapon from Mirkwood.