First of all, it should be noted that the Chinese -did- achieve several of their major objectives during the war (I've written about that here: http://www.reddit.com/r/AskHistorians/comments/339y5g/who_won_the_sinovietnamese_war/cqj02rr). Namely to show the Vietnamese that their Soviet allies would not protect them in the event of a war. China never intended there to be a total defeat of the Vietnamese, in fact Deng Xiaoping (the leader of the PRC) stipulated at the start that the war was to last no more than 32 days. The PLA made inroads into Vietnam and then withdrew largely voluntarily (and by that I mean they were not pushed out of the country by the VPA, but did suffer significant casualties) after a couple of weeks of fighting. Therefore, you cannot portray China's invasion as a total military failure, and that the Chinese did plan before the war to withdraw around the same timeline that they actually did.

Second of all, the PLA had being severely damaged by the culture revolution. The Korean war and the 1962 Sino-Indian war had shown the effectiveness of the PLA militarily. However during the culture revolution, which had ended with Mao's death just 3 years before the war, the PLA's primary objective became that of political indoctrination and training rather than fighting wars. A number of talented officers, including the Marshal who commanded Chinese forces in Korea, were purged. This meant the quality of leadership during the war was particularly low.

During the war despite having superior numbers the PLA was hampered by the lack of logistical capacities, food had trouble getting to the front and were often ferried by peasants rather than the military. The lack of trucks and APCs severely limited mobility. Communication was often done by couriers carrying messages between units rather than radio. The lack of air reconnaissance meant that ground troops took heavy losses when they ran into unexpected obstacles. Combat effectiveness was low relative during the war.

Not only that, China was fighting the war with second-line units, it kept its best units in the north on the Sino-Soviet border in anticipation of possible Soviet retaliation.

This is of course, not to understate the effectiveness of the VPA, which received a significant number of Soviet advisers before the war began. Of course, this army had being battle hardened against ARVN and US forces over the previous decade. By comparison, the last time the PLA engaged in a major ground war was in 1953. This meant a significant gap in the level of combat experience between the two armies.

Basically the PLA which invaded Vietnam was a complete mess, it fought for limited objectives (some of which were achieved), and were suppose to withdraw in any case. China could have chosen to continue to pour manpower into Vietnam and try to take Hanoi, but this would have led to the very sort of quagmire that Deng sought to avoid and greatly increase the possibility of Soviet intervention. Not to mention that China probably would not have succeeded even if it did try. This gives the appearance of a Chinese rout, but in fact (see the post I linked at the beginning), it could easily be argued that the Sino-Vietnamese was a short term Vietnamese victory but a long-term Chinese victory.

This is actually something I am pretty familiar with.

So in 1978 the Chinese military was in a somewhat weird state. Since the 1927, the PLA operated under the doctrine of a "people's war". This doctrine led a bigger emphasis upon defensive strategy and manpower-over-weapons.

"(3) Make wiping out the enemy's effective strength our main objective; do not make holding or seizing a city or place our main objective."

"(4) In every battle, concentrate an absolutely superior force (two, three, four and sometimes even five or six times the enemy's strength), encircle the enemy forces completely, strive to wipe them out thoroughly and do not let any escape from the net."

This mentality hindered the modernization of China's army, even after their development of nuclear weapons in 1964.

There still existed a faction within the Chinese government that desired modernization. The Four Modernization campaigns enacted in 1978 were designed to modernized the country, militarily and domestically, in order to improve national security and their economy; however, it was a bit too late in regards to the Sino-Vietnamese War.

The Sino-Vietnamese war was a "preemptive self-defensive counter-attack". The goals of the war wasn't the complete takeover of Vietnam. It was to be a limited war, only utilizing ground forces. The main goal as Deng Xiaoping said, was to teach Vietnam a lesson to humble them and once the objectives were completed, they would withdraw.

At this point in time, Vietnamese forces had invaded Kampuchea. They also made repeated incursions into China, killing both Chinese soldiers and civilians. Domestically, the Vietnamese had persecuted the Hoas, southern Vietnamese people who were ethnically Chinese, leading to a refugee crisis.

The Vietnamese operated under the belief that their Soviet allies would stave off any major offensive by the Chinese, either through direct military force or through pressure. While the Soviets were a small factor to the Chinese diplomatically, militarily they were largely ineffective at stopping the Chinese offensive.

Comparing military forces, it makes it hard to believe that a force of 4.3 million were stalled for so long by a force of 600 thousand. The PLA ended up sending roughly 330,000 men, 10% of their ground forces, to fight with the VPA which numbered 200,000 to 250,000 on the northern border.

The majority of the PLA's weaponry were outdated. Their anti-tank weapons lacked any advanced firing equipment. Their artillery were limited by their mobility and inaccuracy. Their guns were also archaic, some of which being World War II leftovers. The PLA were severely hindered in the war by transportation and logistical issues.

The VLA were for the most part, well equipped and well trained and were arguably stronger than their counterparts in the PLA; however, they were limited in number.

The PLA was hindered by the terrain of the mountainous border, slowing down their troops. They were forced to rely upon old trucks, horses, and donkeys for transportation and logistics. Initially, the VLA utilized local troops and militiamen at the northern border to stall while their forces assembled further south to prepare for a battle outside of the PLA's artillery's range. The PLA was further hindered by a network of tunnels and trenched manned by the militias. The guerrilla warfare utilized by the Vietnamese during the American Vietnam War was also used to great effect against the Chinese. The PLA were slowed down as they began conducting counter-guerrilla tactics, tunnel searches, dynamite detonation, and mine explosions.

This merely slowed the PLA down, however, and on March 3rd, Lang Son was captured. Lang Son was a strategic city vital to the Chinese. The north of Lang Son was rugged while the south was a open and flat. It was effectively the last line of defense for the Vietnamese as the city would be the launch point of any offensive to Hanoi. The PLA would soon withdraw a few days later, saying that the goals of the war were met, destroying the infrastructure of the North as they went back.

The war exposed the PLA's major weaknesses leading to the reevaluation of the "people's war" leading the conclusion that the PLA must modernize, both strategically and physically. They began to modernize their air and naval powers, their air force being extremely weak in comparison to Vietnam during the war.

Sources:

Chen, King C. China's War against Vietnam, 1979: A Military Analysis. Baltimore: School of Law, U of Maryland, 1983. Print.

Mao, Zedong. The Present Situation and Our Tasks. Peking: Foreign Language, 1961. Print.

Roberts, Priscilla Mary. Behind the Bamboo Curtain: China, Vietnam, and the World beyond Asia. Washington, D.C.: Woodrow Wilson Center, 2006. Print.

Wikipedia contributors. "Four Modernizations." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 31 Mar. 2015. Web. 17 May. 2015.