Unfortunately, because a significant amount of the normalization has occurred within the past 20 years, it falls outside of what we can discuss here. I can, however, point to a few of the underlying dynamics that continue to this day. It takes two to tango, so I'll start with why America generally views Vietnam in a positive light.

First, many liberal Americans viewed the Vietnam war as a strategic mistake and a brutal war, but placed the blame more on domestic political leaders than on the Vietnamese people themselves. This positive perception was aided in conservative circles and beyond by our alliance with South Vietnam during the war, thus helping to curtail or diminish state-sponsored racial animus towards Vietnamese people. Note the difference between the Vietnam War and World War II's Japanese internment camps and Admiral Halsey's declaration that "the only good Jap is a dead Jap."

After the war, televised and dramatic images of the Fall of Saigon and the so-called 'boat people' helped to generate sympathy among Americans. All of this was augmented by Cold War propaganda which frequently portrayed people living in communist countries as oppressed. The resultant, decently sized Vietnamese-American community has influenced and been increasingly accepted into American culture. This community also served to bind the countries not just culturally, but economically, as in the 'doi moi' policy that /u/AGoodIntentionedFool mentioned.

On Vietnam's end, it has mended ties with America because of the necessity of power balancing against China, which has historically been the main threat to Vietnamese sovereignty.

Robert McNamara was the US Secretary of Defense who presided over much of the pre-Nixon escalation of American forces in Vietnam. He had this to say about the conflict:

In the case of Vietnam, we didn't know them well enough to empathize. And there was total misunderstanding as a result. They believed that we had simply replaced the French as a colonial power, and we were seeking to subject South and North Vietnam to our colonial interests, which was absolutely absurd. And we, we saw Vietnam as an element of the Cold War. Not what they saw it as: a civil war.

Thus, even though North Vietnam (and later the whole country) were nominally in the same Cold War 'bloc' as China, they didn't necessarily ally with China for ideological so much as practical reasons. China provided Vietnam with crucial support during the Vietnamese war in return for influence over its government. However, once the war was largely over, tensions between Vietnam and China began to rise as some in Vietnam grew resentful over said influence. They viewed China as one more colonial power to be fought off, returning to the historical dynamic. Indeed, for much of the past thousand years Vietnam (and the states that preceded it) were alternately part of Imperial China, or struggling to be independent of it. (See: the story of the Trung Sisters)

China supported the Khmer Rouge's state and Vietnam militarily intervened against them. During the Sino-Soviet split, Vietnam strategically repositioned itself into an alliance with the Soviet Union, breaking away from China's sphere. From 1979 to 1990 it and China carried out a persistently-low-grade, occasionally-punctuated border war, which tends to sour relations. It has remained relatively self-contained and unaligned since the fall of the USSR (militarily, economically, etc.).

I don't want to break the 20-year rule, but look at the recent headlines surrounding the South China Sea, and draw your own conclusions about any recent changes in US-Vietnam relations.

Edit: /u/DBHT14 's comment