In what ways did the Virginia Plan and the New Jersey Plan demonstrate the difficulty of creating a new nation?

The Virginia Plan and the New Jersey Plan were competing suggestions presented during the Constitutional Convention in 1787. In the years following the Revolution, the Articles of Confederation had been the governing principle of the young nation, but it was quickly determined that the Articles were insufficient. Specifically, they did not offer guidance as to how to resolve conflicts between the states nor did they address the issue of how each state could be adequately and fairly represented in the federal government. The Virginia and New Jersey Plans sought to resolve these issues, but the fact that they largely ran contrary to each other gives some indication of the sorts of problems the nation faced at this time.
Virginia was a large state with a large population. Accordingly, it is not surprising that the Virginia Plan, based on the suggestions of James Madison, seemed to favor larger states. It favored a two-house system with the upper house elected by the lower house; the lower house would be elected by the people, who were to be represented proportionally. This meant that a state with a high number of free inhabitants would have more representatives in the lower house, and by extension, more power over who was elected to the upper house.
The New Jersey Plan represented the concerns of small states like New Jersey. Its proponents rejected the suggestion that larger states should have more control. Under the New Jersey plan, each state would be represented equally with one vote each, as had been the case under the Articles of Confederation. This view reflected an understanding that each state remained independent; a state, as an independent entity, would be represented in Congress, whereas under the Virginia Plan, each person, as a citizen of the nation, would be proportionally represented.
The question of states' rights caused a huge amount of strife in the early years of the Republic, which is part of why the Constitutional Convention happened in the first place. The Virginia Plan and the New Jersey Plan epitomized this conflict. Ultimately, the Connecticut Compromise drove what was finally decided, a compromise between the two plans that strove to represent both large states and small. The proportional representation argued for in the Virginia Plan was indeed granted in the House of Representatives. In the Senate, however, each state would have one vote, as the New Jersey Plan called for. This compromise was an attempt to acknowledge that every American citizen should be federally represented, as citizens of a nation, and that larger states should not be "more important" than smaller states.
https://courses.lumenlearning.com/boundless-politicalscience/chapter/the-constitutional-convention/