I think, after reading what´s written above, we must make a distinction between systems that are big and little, like the atmosphere and a single billiard ball.
I´s clear that a billiard ball on a table makes a journey that deviates more and more from the path it would have taken hadn´t you give a slightly different direction in velocity.
Consider now a container with a gas consisting of all the same atoms, and consider these atoms as billiard balls (I think not invoking quantum mechanics is reasonable assumption). In the case of the billiard ball, the system consisted of just óne element (of course the ball consists of a lot of tightly connected particles, but for the ball as a whole they´re not important; how often do you hear to consider atoms as billiard balls?). The trajectory of an atom is surely gonna be different if the direction of this one atom gets a slightly different direction. But this time, there are no walls of a table. It bumps into other atoms who on their turn get also a slightly less velocity direction.
The number of atoms is so big that after a time the different direction in velocity of the first atom won´t be noticeable anymore for most atoms of the gas. So the whole gas-container system won´t change in comparison to a container (of course with the same dimensions) that contains the same gas. And even if we should change the direction of the velocity of each atom, there would be no macroscopic difference concerning the two boxes. Off course, all the atoms velocities would differ, but for the system as a whole, there´s no observable differences. Same pressure, temperature pression, etc.
Now the example of the boxes with gas is rather simple (but of course you can´t keep track of the individual atoms) because the gas is contained. But now imagine a river. If I change the direction of velocity of one water molecule somewhere high in the river, the molecule, of course, changes its path, along with many others (direct or indirect). But these changes dampen out and the river ends up the same as without these little change of one molecule.
But if the velocities of the molecules of the river as a whole (all the water molecules) each get a little different velocity direction, the river sure is gonna end up different down the mountain.It ends always down the mountain, but there are many ways. The difference with the container of gas is very simple: the gas is contained, the river runs wild, without container boundaries.
The same is true for the atmosphere. Change one little part, and there will develop no different weather (maybe some little turbulences around the butterfly). Change all the molecules that make up the atmosphere (conserving the energy), and a completely different weather develops.
I read once that if you connect the butterfly to a very sensitive atomic device (by means of a radio transmitter tied to the butterfly) and if the butterfly makes a slightly different movement so that it gets close enough to the bomb so that it will detonate (gee, how far have we come?), the little change in movements of the butterfly has a very great impact on a large scale. But I think in chaos theory we must consider systems that have no energy load they can trigger, wich in fact, the butterfly also has, but in this case, it´s a very little energy that the butterfly possesses, and she is part of the surroundings, not of the weather itself (the explosion of an atom bomb would also be part of the boundary, but the amount energy involved in a butterfly is negligible). Or take the story of a man who missed his plane, because he júst didn´t run too hard to catch it. Unlike an atom in a box who júst didn´t hit another atom his trajectory amidst the surrounding, won't cause a damping of the things to come (because of his energy resources). He´s pissed of, decides to go back home (he had an important meeting in London, concerning a big money deal), and angry as he is, drives much too fast on the highway, after wich he enters town and still puts the foot on the gas pedal. He doesn't see the woman and child, who both get killed.
Obviously, the last examples are no examples of chaos because parts of the systems have their own internal energy source.