Important governmental powers not held by the Articles of Confederation Governmnet: A good book for you to read is The Articles of Confederation by Merrill Jensen. Is is not too long. It has been years since I read it, so my answer will not be so good as you might get from others, but here goes.
Given that the USA was governed by the Articles of Confederation throughout the long War of American Independence, and given that the USA won its war, there may have been no essential powers lacking, but your question was about important, not essential, powers. Under the Articles, the Confederation government could only ask the states for money; it could not take taxes directly from the people. Being able to directly tax the people for money to finance a defense establishment and to finance international diplomacy, permits a better army and navy and more respect in international relations. Direct taxation has its drawbacks. One example is "The Road to Nowhere" built in my county by the U.S. Forest Service. The road goes for several miles through the mountains; it is a wide 2-lane paved road; it ends at a 1-lane, county, dirt road. If the dirt road is good enough for residents to drive on every day, why is "The Road to NoWhere" necessary for hauling logs only part of the year? Answer: It isn't necessary, but government bureaucrats like to spend money; it makes them feel important and keeps them busy. Direct taxation makes it too easy for Congress to spend money all over USA on pork-barrel projects. Perhaps the government should be allowed to lay direct taxes only for defense and diplomacy; perhaps all other federal monies should have to come from requisitions on the states, as it was under the Articles.
Another problem under the Articles was that many, maybe all, states had regulations that restricted trade. For example, if a state had a hat-making industry, it might prohibit the importation of hats from other states. This prevented consumers from purchasing the best hats for their needs; it prevented the best hat makers from making the profit that they deserved; it permitted the local hat makers to charge higher prices than they deserved, and to sell poor quality hats. Some amount of control of interstate commerce was needed to stop these sorts of abuses. The Articles gave the Confederation government no control over interstate commerce. Under the present Constitution, the federal government controls interstate commerce, and there have been abuses of that power by both government and special interests. Some amount of control is better than no control; too much power to control may be worse than no control.
The reason the Articles did not give the Confederation government all of the powers that most governments have, is that the people setting it up saw the state governments as the legitimate governments and the Confederation government as only a means to do together the few things that the state governments could not do individually. The people who wrote the Articles had just seceeded from their home government located in London, because it had tried to use its power to lay taxes and regulate commerce in ways that many Americans believed to be unjust. They did not want to give these same powers to the government that replaced the one against which they were rebelling.
The Articles of Confederation also required 9 of the 13 states to vote yes in order to pass a national law, a supermajority that was difficult to achieve in ideal circumstances since the colonies were so different in culture and opinions. In order to amend the Articles themselves, a unanimous 13 of 13 states was required, which was quite impossible. So as the government was weak and broke, it also did not contain the means with which to fix itself.
The idea behind the Articles was to establish a decentralized government - essentially the opposite of a King. It worked pretty well during the revolution since it was an emergency, and each state kicked in its share out of need for survival. Once the war ended and the emergency was over, states started acting in their own self interest first, rather than the nation's.
Coming out of the American Revolution, the Articles of Confederation created a very weak national government. The fear that still loomed in the minds of the Framers was that a situation such as England and King George, which represented the essence of a strong central government, would still arise in American politics if the centralized form of government had become too strong. This led them to design a weak national government under the Articles, one that could not tax states without the states' consent. The federal government could not also negotiate foreign treaties without agreement from the states, and contained no provision for federal courts.
The national government under the Articles of Confederation was simply too weak. It lacked an executive (President) and it lacked the ability to tax. It could not force or require the states to do anything (like pay for the army to fight the Revolutionary War).
The reason that the colonists set up their government in this way is that they were overreacting to their experience with England. They felt that Great Britain's central government was too strong and that that was the root of the colonies' problems. So they decided to make a system that was diametrically opposed to the British system.