It all adds up to easier riding, which translates into a less tired biker at the end of the day. A light bike is not everything, though. A wide selection of gears will also make your day more pleasant. Lastly, a bicycle that fits will be infinitely more comfortable than one that doesn't. If you can't afford a new bicycle, don't despair.
People ride the tour successfully on bicycles of every description.
Do take the time to get whatever you are going to ride in top mechanical condition though. You probably know by now that when you buy your bicycle, you aren't done spending money yet. The most important accessory is a good helmet. They aren't expensive anymore, certainly less so than a lifetime of invalid care. You MUST wear it on Cycling the Erie Canal, but you'd be crazy to leave it home at any time.