I know a lot more about Korea than about Vietnam but I'll try to hit both sides:

Vietnam

North Vietnam won but battlefield victories don't guarantee economic prosperity. Quite the opposite in the case of Vietnam. Having to fight two long drawn out wars (three if you count WW II) that involved hugely destructive bombing campaigns is not good for your economy. The Vietnam War (including fighting against the French) took a very long time and it really ground the country down in ways that the Korean War, however horribly bloody it was, just didn't last long enough to do. Because of this the Vietnam War ended quite a bit later than the Korean War so Vietnam has had less time to recover.

Unifying a country also doesn't necessarily bring in economic prosperity. Did West and East Germany unifying lead to an economic boom? There are some benefits in terms of economies of scale to unifying a country and a lot of people like it for nationalistic reasons but unifying a country and integrating two very different economic systems is very expensive. A lot of South Koreans really don't want to unify with the North because of how expensive it will surely be.

Finally I'm not sure what "some reform" you're talking about. After a while North Vietnam moved away from Soviet-style systems of economic organization but I'm not seeing anything here that would give Vietnam an edge over Korea.

Korea

The Korean War was horribly destructive and it turned a lot of Seoul into rubble. However, a lot of the destruction of the war was in the North (due to large bombing campaigns) and a lot of the areas south of Seoul only saw fighting in the first part of the war and the war ended a long time ago so Korea has had a longer time to recover than Vietnam.

Also the Vietnam War itself was hugely profitable for South Korea (much like the Korean War was hugely profitable for Japan). As an Asian ally of the US, America pumped a lot of money into Korea. A lot of war supplies were purchased in Korea by the US and the US paid the difference between the wages of American soldiers in Vietnam and Korean soldiers in Vietnam (most of which was pocketed by the South Korean government instead of paid to the Korean soldiers fighting in Vietnam because that's the sort of things dictators do).

Also the US opened its economy up to South Korean exports. For quite some time things like t-shirts, various knick-knacks, and plastic toys were made cheaply in South Korea and then shipped to America. A lot of the same sort of low value manufactured items that came to be associated with "made in China" later on were made in Korea when I was a kid in the 80's. This kind of large export market allowed Korea to build up to the point where they could successfully make the transition to more expensive goods later on.

Vietnam didn't have any kind of large open export market like that. Vietnam was on good terms with the USSR but the amount of trade it could do with the Soviets was limited (and even there South Korea had surprisingly warm relations with the Soviets, used to tutor a guy who was an Assistant Minister of Agriculture for South Korea in the 80's and he liked to go on long rambling stories about how well he was treated during government trips to Russia). Meanwhile Vietnam intervened in Cambodia (because of Pol Pot) and was invaded by China in a hilarious incompetent military operation.

Meanwhile the various oil crises of the 70's were also good for Korea despite Korea being a major oil importer. South Korea has long had good relations with a lot of OPEC countries. When various Middle Eastern companies wanted to pay for large infrastructure projects they often didn't want to contract with Western countries (because of support for Israel) or communist countries (because they hated communism) so Korean companies got a lot of business building infrastructure in the Middle East and were able to do it cheaply since the Korean workers they went over had very low wages in those days. A lot of Korean companies still have a lot of profitable business contacts in the region today they just get a lot of the labor from South Asia instead.

While Korea was still poor and developing a lot of South Koreans moved abroad. Often to the US or to West Germany. A lot of these people returned home with some money in their pockets (the Koreans who went to West Germany generally had short term guest worker visas and didn't stay) or at least sent money home. This wasn't the case as much with Vietnamese immigrants as they were generally people who hated the Vietnamese government and didn't want anything to do with it.

So South Korea had a lot of very profitable international contacts while Vietnam had Pol Pot being a monster and the Chinese invading them. Really stark difference there. In recent years Vietnam has seen some strong economic growth moving into the same kind of low end manufacturing economic niche that Korea used to be in as a lot of that kind of thing has started to flow out of China for various reasons. But Korea has already moved out of that niche a long time ago so it has a big head start on Vietnam. Vietnam is also doing well with tourism (until COVID-19 of course), notably with South Korean tourists loving Da Nang.

South Korea also had some luck with its dictators. The three main dictators who ruled South Korea for the bulk of the time before democratization were Syngman Rhee (senile and incompetent piece of shit), Park Chunghee (a pseudo-fascist monster, but very lucky and cunning), and Chun Doo-hwan (horribly corrupt and brutal piece of shit). They were all horrible dictators but South Korea got lucky that it got the smartest one at just the right time who was cunning enough to suck up well to the US and get a lot of economic benefits from that without alienating countries that didn't like the US much. A good example of him being smart was how he negotiated a treaty to normalize relations with Japan. Japan paid Korea a big stack of money in reparations, comfort women compensation, etc. Park was a pretty sharp negotiator and was able to get $800 million in grants and loans, which was quite a lot of money at the time. Rather than giving the bulk of the money to the victims of the Japanese in Korea he stiffed the comfort women etc. because he was a huge piece of shit and used it to fund various government projects.

His corruption was also more skimming some off the top than bleeding the country dry as many other dictators in the developing world did. A lot of his economic policies still cause problems to this day, such as massively favoring huge corporations and certain regions of the country over others but he was smart enough to not fuck up the economic boom that got going under his rule. He also had a string of good luck. For example he started building a ludicrously expensive (compared to the very weak Korea economy at the time) Seoul to Busan highway that looked like it could've pushed Korea into a debt trap but then right when the loans were coming due the economy boomed and he could service the loans.

Then after he got shot by his own spy chief, Korea's economic growth had enough momentum that it was able to survive Chun Doo-hwan being a piece of shit without any redeeming features.

After Chun Doo-hwan, Korea was able to transition to democracy which has not happened in Vietnam. Three decades of democracy makes a huge difference when it comes to economic development. You don't see a lot of rich countries that aren't democracies (unless they hit oil or have a lot of luck) and Vietnam is very much not a democracy while Korean democracy for all of its faults is quite strong. Presidents even get kicked out and jailed when they commit crimes which doesn't happen in most other countries.