The character of peace treaties is only partially related to the character of the following peace, and to the inevitable end of that peace. It is not possible to draw a direct line from Versailles to September 1939. Certain events had to happen in the intervening period for World War II to start, including the Great Depression, and Britain and France's basic abandonment of maintaining the peace in the face of adverse pressures. The basic sin of (particularly British) statesmen during the interwar period was that they believed that no one could want war or find utility in it, and that peace was self-sustaining. History proves that neither assumption was true. Britain and France, the two main peacekeepers with the withdrawal of the US back into relative isolation following WW1, had by the 1930s become unwilling to enforce the peace, believing that piously liberal words would suffice.

What was (and has been) the character of the peace following the Second World War? For at least its first forty years, peace was enforced at gunpoint. Notwithstanding (or perhaps because of) the horrific danger of a superpower nuclear conflict, Europe was frozen stable (mostly). NATO was founded, largely through British initiative, to "keep the Americans in, the Soviets out, and the Germans down". Basically, to maintain the peace that had been so dearly won. Innovations such as the European Coal & Steel Community also played their part, in part by forging a new interdependence between the nations of western Europe but also by giving non-Germans some basic oversight and power over the exploitation of German resources. Sort of like how Versailles gave the French oversight over the Saarland and other important bits of Germany. The difference is that the second time around, the British and French were actually serious about it, and had managed to rope in the Americans as well.

Basically, peace is not self-sustaining. Interdependence alone is hardly enough to dissuade nations from war or other costly policies. See how the First World War started. Or see Russian policy in the past year. Basically, during the interwar period, none of the status quo powers seriously sought to maintain the peace after the first few years (the joint Franco-Belgian occupation of the Saar in 1924 was IIRC the last time a western European country seriously sought to enforce Versailles). After the Second World War, the status quo powers seriously sought to sustain it, and have thus far largely succeeded in doing so.

Sources: Williamson Murray, "Versailles: the peace without a chance" in Murray & Lacey (eds), The Making of Peace; Murray, "British Grand Strategy, 1933-1942" in Murray, Sinnreich, & Lacey, The Shaping of Grand Strategy; Donald Kagan, "The End of Wars as the Basis for a Lasting Peace", Naval War College Review.

Kasirzin's answer is great so I'll just add a little something. The Myth of the "Stab in the Back" was able to be perpetuated was because at the signing of the armistice, Germany had not suffered total defeat like it did in WW2. German armies still "held" much of France and would take months to recall to Germany.

(Note: Total defeat defined as the complete destruction of your armed forces, occupation of your country by a foreign power and no ability to make independent political decisions)

It's interesting you specifically point to peace between the former axis and allies, since it wasn't an entirely peaceful era with regards to the west vs ussr. i would argue that there weren't treaties in the conventional ww1-like sense after ww2, as both germany and japan were completely occupied. this occupation (germany was split, japan ruled by the americans) led to the ruling powers being able to coerce friendly govts and policies, and tie these countries to their sphere of influence. both were also demilitarized and stationed thousands of allied (american and british) troops - which interestingly acted as both an occupying force of sorts (to ensure that neither japan nor west germany has any ideas about attacking the victors) but also as a protector force (genius!). so successful has this been that to this day tends of thousands of american troops still are stationed in both.

also the rise of communism and the ussr meant that these former enemies became much more useful as new allies in the new fight, and investment and rebuilding them as bulwarks against communism was the priority, not pillaging them for resources. i'd argue that was the real reason of success, not treaties meant to tie interdependence, but occupational power that led to the west dictating policy to these countries, and the threat of communism which led to forcing these countries to be allies rather than punishing them.

One important aspect is that the two global super powers that arose on the periphery of Europe (the US and UDSSR) projected their power into central Europe. After WWI both the winners and losers were very weak. The one power that had benefited from the war, the US, withdraw from Europe back into isolation. France and the UK nominally reached their peak power, but were not able and willing to defend their status. Russia went through revolution and civil war. Germany collapsed and multiple new nation staates appeared on the European map. The political system resulting from all of this was weak. Many states were not stable to begin with, new movements (communism, fascism...) destabilized it further and the economical crisis washed away what was left of confidence in the status quo.

So when Germany became aggressive in turning back the results of the Versailles treaty neither Britain nor France was willing to commit to enforcing them. This lead to WWII.

Now after the second world War, Germany and France were done as world powers. britain became the junior partner to America and the Soviet Union became a global leader. Both super powers created spheres of influence in Europe they were willing to protect and secure at any cost. Resulting in an uneasy but very stable cold war.

By "peace", you mean the Cold War?

You should be asking, it would be more accurae "why didn't the Cold War turn hot?"