Because the Vietnam War was an insurgency, or rebellion, it required a completely different kind of fighting. The United States Congress has not declared war since December 8, 1941, when it did so against Japan. Ever since then, the President has ordered troops into harm's way almost at their sole discretion. Congress' only limit is that they can eventually deny funding for the war, but this is almost always politically unpopular because it appears in some way unpatriotic.
Also keep in mind that Nixon was President for the second half of the Vietnam War, and his corruption extended into many areas, including his carrying out a secret bombing campaign in Laos and Cambodia and special forces incursions into those areas without Congressional knowledge or approval. This alarmed them as anti-war sentiment was brewing at the time, and they worried about the safety of their seats, especially if they were in Nixon's Republican Party at the time.
This concerned Congress because Congress and the President are, in a sense, in competition. The Constitution gives each "side" some of the power over war and neither side wants the other to take "too much."
So, when the President started taking more power, the Congress became worried. You could say this was out of a selfish desire for power or you can say it is because of principle. The principle of the issue is that power was split between the two so that no one branch could go to war without the consent of the other.
As things stand now, the President can kind of go to war without Congress approving it. This worries some people who think that is not what the Founders wanted. This was one reason why Congress was alarmed.