Awesome question! The fact is, there are a lot of misconceptions about the Civil War, and the Confederacy itself. Had the Confederates managed to hold out long enough for the war to be untenable for the North, we would have seen some major changes - and surprisingly, a lot of things would be the same.
First of all, you can't address this scenario without facing the elephant in the room, so let's get it out in the open: Slavery. True, it was positioned as a major factor in the war, but it really had little to do with what caused the conflict. Regardless of who won, pressure was being put on the South by abolitionists on both sides, as well as their only true ally in the conflict, the British. Within a decade, the slaves would have been freed.
As for the Confederacy as a nation, there would be a dramatic increase in revenue, as the northern states were still dependent upon them for agricultural crops such as cotton, tobacco, soy, and others. The main difference would be that the Confederated States would be able to sell those products without the price ceiling, tax manipulation, and quality determination previously imposed on them. The Southern states would also be able to sell their goods to foreign markets - an option blockaded by merchants in the north.
It is anyone's guess how the expansion in the West would have been effected, but I suspect the machinations for that expansion were already well into play, and it would have continued on regardless.
It is a certainty that the outcome of World War One would have been effected, but it is arguable as to what extent, since the United States entered into the war at a fairly late date. It could even be argued that, without the United States the division of Europe would have been so greatly altered as to make World War Two non-existent.
In the end, what we would see at the end of a hundred and fifty odd years would be a world that would be far less technologically advanced, but perhaps a bit more peaceful.