@pdworkin, @laureth, and @Jeruba have it right. Money is a convenient method of trade.
It seems to me that you have an idealized belief of a greedless society that would result by not having money. This is not so.
If money was eliminated, that would not change human nature. That would not mean that wealth and resources would be distributed amongst the others. Wealth would be accrued the old way, in such as precious metals, gems, or whatever is of value in your society and hoarded by individuals… just in a bulkier form. Money is not the root of all evil. Greed is not dependent on money.
Even if all money was distributed amongst the poor, I do not believe that would end poverty, famine, and the like. This planet is just too overcrowded for that.
Ending illness? No amount of money or distribution of resources is going to end illness. Diseases give the rich the finger all the time, taking them down fast and hard. Yes, the poor die from illness much more frequently than the rich, however, there is no end to illness.
War and crime would not be ended if money was eliminated. Crime does not always have to do with actual money and war is usually not fought due to money. Yes, sometimes it is, but usually not.