Most of the others have really covered it. Often it isn't a question of if you need a better bike, if the one you have works fine and has nothing wrong with it then the obvious answer is you don't need a better bike. Unless there is something wrong with your bike that keeps it from working properly the real question is do you want a better bike?
A better bike will work more efficiently so while it won't "make you faster" it will better help you better realize your potential which of course means faster results. More importantly is that a better bike often stops better which I find pretty important. I know people got by fine for years, and still do, with hard little blocks of rubber on chromed steel rims using a less efficient leverage design and softer, thinner metals to stop but I'll take a modern dual pivot or disc brake any day and the results can make a serious difference in the right situation.
As to used, too many people recommend this but if you don't know what you're looking at used isn't always more affordable and can be a money suck.