Yeah, the width of the blade is rather `all-enveloping`, isn't it? I cannot really remember which source gave that data, but I've read others in which the blade should be between 1.5 and 2 inches. There are differences however, which I'll try to explain here:
1. The main difference between rapiers and longswords is in the design, however. Rapiers have blades that get narrower as you get closer to the point, while longsword blades stay rather wide all the way until the tip of the blade where it gets pointy rather quickly. Ok, I don't know how well that describes it, but let me try some ASCII art here:
Rapier:
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Longsword:
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===========================--------
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2. The other difference is the handle. Longswords usually have only the crossguard (or quillions) to protect the hand, and they are rather short. In rapiers, however, the quillions are long, almost ridicilously so in some instances. The (short) handles also include finger guards (in front of the quillions) and knuckle guards. Later, rapier handles became `so-called` swept hilt designs and even later `so-called` `cup-hits` appeared.
3. The way you hold the blade in your hand is very different. You hold the longsword as if it was a broadsword, which is almost like you were holding an axe (fist grip). When you hold a rapier, you have at least your forefinger crossing over the quillions - sometimes both forefinger and middle finger. In the case of a single finger crossing the quillions, it is places underneath the blade to support your grip. The fingerguard is then only on that side of the blade. In the case of the `two-finger` crossing, your index finger is on top of the blade and middle finger is under the blade.
4. The way you use it. Longswords are mainly hacking weapons and only a slight development from broadswords. You could hit the opponent with the edge up to the very end of the blade without fearing that it would break. Rapiers, with the grip I just described, are more 'finesse' weapons -> you get more exact control over the point of the blade with a grip like that. and it works better for thrusting. You could only use the edge for the first third of the blade for hacking - if you used the point section, the blade would easily break.
In my sig you see a small picture of a rapier typical to the 16th century and early 17th century. The handle has a simple knuckle guard and there is a finger guard in front of the quillions. The blade is rather short (too short for many later swordsmasters). In the pic the blade appears perhaps a hair too thin for the period (I blame the resizing).