Read this. This is one of the best things ive read on the similarities and differences. they aren't the same sport, but they are very similar.
Questions like this are difficult to answer, because they usually indicate that the asker is basically unfamiliar with one of the sports involved in the comparison. First of all rugby players do wear pads -- just not the rigid plastic sort that gridiron players use. The rule in rugby is that the material must be soft, and no more than 1 cm uncompressed. You can find all sorts of thigh, shoulder, kidney and chest protectors available for rugby players within that rule. A lot of us (particularly the forwards) wear shin pads as well.
First of all there are very few substitutions allowed in rugby -- you dress 23 with 15 starters all substitutions are generally permanent. This means that at least 7 of starting players must be on the pitch for the entire 80 minutes. There are a lot fewer breaks in play as well. Believe me, after 75 minutes of running up and down the pitch and when you are into your seventh or eight mile, you are going to notice the extra weight from those rigid pads, and they are going to be interfering with your ability to run as well. You are also going to be having a lot of difficulty losing heat -- particularly if you are wearing an American style helmet. Heat stroke would be a real risk.
It is also important to realize that the nature and purpose of tackling (and more importantly rucking) in rugby is fundamentally different than in gridiron. In gridiron you want to stop the runner cold -- to not give up a single inch more than necessary -- because of the downs for distance system and the uncontested breakdown. A few inches can mean the difference between keeping or losing the ball So you hit the runner as hard as possible in order to kill his momentum. The position you end up in after the hit doesn't really matter, because once the runner is down, the ball is dead.
Not so in rugby. After the tackle the ball is still live -- a player off his feet must release the ball and make it available for some other player. As a result, the purpose of the tackle in that rugby is not so much to stop the runner (albeit that is generally a good thing) but to place the tackler in a position where he can remain on his feet and poach the ball. If you can flip the runner so that the ball is on the wrong side of his body, for the ruck so much the better. Similarly, the purpose of the ruck is to make the ball available to a teammate. Either way, you don't (in most circumstances) want to commit to the all-out charge or hit because it means that you can't stay on your feet and compete for the ball.
The laws are different too -- in rugby you can't lead with your head, leave your feet, or shoulder charge. You must make some attempt to "wrap" up the runner with your arms when making the tackle. Again, it makes the rigid padding less necessary.
One should also consider that the padding and especially the helmets may actually add to the danger as players (with a false sense of security) will tend to hit recklessly. Rugby certainly doesn't contend with the degree of deaths, crippling injuries and concussions (and early onset dementia that follows) that gridiron players experience.