What did Wilson say were the aims of the war and how do those compare with why the US really entered the war?

The United States, an exporting and a dominantly-industrial nation at the time, did not enter the First World War by the same means as their allies did, as the U.S. saw the war as an opportunity to prove (and expand) their sphere of influence and industrial power. At the start of the war, President Wilson himself assured the people of the U.S. that they -responsibly - would not be not be joining the war, and that they would pose as a neutral.
However, there were also other triggers that prompted the US to enter WWI. Germany had promised to liberate the seas with unrestricted submarine warfare. Yet, Germany broke this promise by unleashing their fleet of U-boats, which could sink a ship within minutes of firing a missile. On May 7th, 1915, one of the German U-boats launched a torpedo towards a British ship which held more than 120 Americans on the boat. Although Germany ended up apologizing for sinking the ship, they justified their actions by pointing out that the Lusitania was loaded with more than 150 tons of British military weapons.
Additionally, nearing when the U.S. joined WWI (1917), the U.S. broke off relations with Germany when it was becoming increasingly visible that Germany was advancing in the war and setting themselves up for a victory. Shortly after, Germany sunk an American ship, named the Housatonic.
Seeing that they were still being involved with nations involved in the war, it was practically inevitable for the U.S. to ignore that they were being tempted into war. While there were also many other economic factors between the U.S. and its allies, the biggest factors of the U.S. entering WWI lie within the straining of their relationship and trust with Germany.
On the other hand, Wilson's aims for the war shifted after the U.S. became completely involved in the war (even though the war was not on U.S. land, but across seas). Nearing the end of the war, Wilson announced his famous Fourteen Points to the U.S. Congress, which begged for peaceful corporation and lasting peace between all involved nations of the war. Thus, also proving that the U.S. mainly came in the war to end it all.

Wilson stated that the United States entered the war to make the world safe for democracy. Wilson did not want the United States listed as an ally but as an associated power, meaning that the United States had higher motives rather than territorial expansion. He claimed that by having these higher motives the United States could provide an example to the warring nations of Europe.
The United States went to war in 1917 in order to punish Germany for its use of unrestricted submarine warfare and the Zimmerman note, a proposed German-Mexican agreement which would possibly lead to another Mexican War. The United States went to war in 1917 to defend its right to trade with whomever it wished. The United States also went to war in order to assist the Entente nations who were dependent on American loans to continue the war. If the Entente lost the war, then they would not be able to repay American financiers. The United States claimed high principles for joining WWI, but money played a key role in the decision to join the conflict.