We did use bombs. We used an unbelievable number of bombs. We dropped more tons of explosives on Vietnam (and Laos and Cambodia) than the total dropped by all combatants in WWII. Nuclear weapons are another thing though; we couldn't use nukes in anger without being pretty sure the Soviets would use theirs on us in retaliation. Nuclear politics is really strange stuff, but the basic upshot is that an atomic weapon is much more a tool for diplomacy than it is a weapon intended to be used.

The problem is this: for all the millions of tons of ordnance we used in Vietnam, bombs can't hold ground. If you want control of a place, eventually you're going to need to send infantry there to physically occupy it. You can make the grunts' job easier by bombing and shelling and establishing air superiority, but there's no way around the need to send them there in the first place. To answer your question more directly, we can't fight without men. There are problems involved in fighting a war that can't be solved technologically. At some point you have to put boots on the ground.

On the nuclear question: The possibility of using nuclear weapons on Vietnam was discussed at high levels, and was part of the popular discussion of the war as well. The military was not necessarily opposed to using them and I believe weapons were deployed to the region (I would have to check this, no time at the moment). But the civilian leadership was never in favor of it. Ultimately there were numerous reasons that resulted in not using them.

One of the major ones is that nuclear weapons are just not that useful in a situation like Vietnam. Nuclear weapons are best used on very built-up targets, or large conventional forces (like battalions of Soviet tanks) — not diffuse guerrilla campaigns or isolated hamlets. There were very few targets where nuclear weapons would make any sense, and those targets were subject to ruinous conventional bombing anyway.

But don't nukes just result in instant surrender? What about World War II? Even with WWII it isn't clear that the nukes resulted in the surrender, and the context was very, very different. Given that the North Vietnamese did not surrender in the face of completely, total conventional bombing, it seems unlikely that nuclear weapons would change that very much. Could they have used nuclear weapons to completely exterminate North Vietnam? Sure. But to what end? How would that further US interests in the region? (And even then, it would be harder than one might think. The largest US nuclear weapons at the time was 25 megatons. That's pretty damned big. But it would take quite a lot of them to really reduce all of North Vietnam to moon-like conditions. Play around with the NUKEMAP if you want to get a sense of how many it would take.)

So the military practicality isn't high. But even more to the point, the diplomatic consequences would have been harsh. The US was not just fighting a war in Vietnam — it was trying to gain and keep influence with many allies and on-the-fence states across the globe. It needed UN support for many of its international goals. To use nuclear weapons in South Asia in the 1960s would have been met with an unprecedented negative response from US allies. These are allies that were already not very happy with Vietnam as it was. UK, French, West German, and Japanese support for the US cause would have taken huge hits if not collapsed altogether. The Soviets and Chinese would be given a carte blanche for doing similar things in their own future wars against non-nuclear powers. There is no way that US nuclear weapons would aid the US diplomatically, even if they did somehow result in a quick Vietnamese capitulation. The "nuclear taboo" was already well established by the 1960s, and even the routine testing of nuclear weapons was digging into US credibility and international prestige.

This, not deterrence, is why the US didn't use nuclear weapons in Vietnam. There was likely no advantage militarily or diplomatically from using them, and the potential disadvantages were huge. I don't think the US really thought that the USSR was going to retaliate with nuclear weapons (the USSR truly did not care that much about Vietnam — it wasn't going to lose its whole everything over one small Asian country, and the USSR was very out-matched against the US in terms of nuclear capabilities until the 1970s) much less China (which had no means of threatening the continental United States until the mid-1970s, if then).

This is an important and subtle point, because there are many other reasons other than nuclear deterrence for why the United States (and other countries) have not used nuclear weapons in anger after World War II. Deterrence explains, at most, why we don't get into direct confrontations with other nuclear-armed states. It does not explain why we don't use nukes against non-nuclear states.

Nixon, as an aside, did try to bluff about his willingness to use nukes, as part of his "madman theory" of diplomacy (if you look crazy maybe people will try harder to make you happy — yes, he did believe this). But I don't think he ever seriously considered using them, and the "madman" approach didn't help his efforts regarding Vietnam.

The U.S. did use conventional missiles and bombs during the Cold War, just not nuclear bombs. During the Korean War, the USAF bombed North Korea with incendiary bombs and conventional explosives for three years up until the armistice agreement in 1953. Over thousands of North Korean civilians were killed in these raids. U.S. warplanes dropped more bombs on North Korea than they did during the whole Pacific campaign of World War II. In Vietnam, the USAF bombed North Vietnam into submission until 1972.

The only reason why the U.S. didn't use nuclear bombs during this era because other countries were developing them too. Also, the U.S. were concerned about our image because world public opinion were still mixed about the dropping of the nuclear bombs on Japan in World War II which ended the war.

Since the Soviet Union detonated its first nuclear bomb in 1949, we realized that we were not the only country to have the monopoly on nuclear weapons anymore. Since then, nuclear arms race intensified with countries competing each other who's got the deadly bomb or not. This means that if we ever get into World War III, nuclear weapons would have been used on a large scale and would make the world a much more deadly place than ever. Just imagine the radiation that would make certain areas inhabitable for years after a nuclear weapon is fired. Therefore, nations decided to use nuclear weapons as a deterrent in order to protect themselves. Ironically, nuclear deterrent is the reason why there is more stabilization of world peace than the pre-1940s wars.