There are a couple of questions here all wrapped into one. So going chronologically:

Treaty of Versailles:

Following World War One, the Entente Cordiale (including the United States, or the "Allied" powers) forced Germany to sign a really harsh peace treaty. Militarily, the Treaty forced Germany to turn over many of its most advanced weapons designs and reduced its army from several million to 100,000 soldiers. They were forbidden from creating an air force, tanks, or any sort of heavy artillery. Their navy was reduced to two obsolete pre-Dreadnought battleships, and many of the soldiers who had gained experience in World War One were sent home. Essentially, the new German Army, called the Reichswehr, was a glorified national police force with no ability to fight even the weakest of Germany's neighbors. For comparison, prior to World War One, the Reichswehr was about the size of the Belgian Army pre-World War One, but lacked the heavy weapons which armies needed to fight a modern war. Now, the Reichswehr used many shenannigans to slow its brain drain, and even created new weapons systems in banned areas. But many of its restrictions really didnt come off until 1933 when the Nazis came to power. The Nazis created a new army, the Wehrmacht, which replaced the Reichswehr. This new army completely flouted the limitations of Versailles, and became the German army we think of when we think of World War Two. How it did that is a pretty well covered story, and is perhaps beyond the scope of your question.

Following World War Two:

Following the Second World War the Wehrmacht was officially disbanded and Germany, at that time divided into four administrative zones, had no army. All its soldiers went home, and the Allies occupied the entire country. But a peace treaty was harder to create thanks to increasing tensions between Eastern and Western power blocs. Rather than create a new Versailles-style settlement, the US and the USSR put the German question aside and instead focused on fighting each other.

Fast forward to 1950 and a lot of funny stuff happened. First, and most importantly, North Korea invaded South Korea. South Korea had a small national army which was supported by a few US troops who were in the process of withdrawing to Japan. They were overrun by a well equipped, fast moving Communist army which ran on Soviet surplus. Planners in the US and Europe feared that the conflict in Korea was a test case for a new Soviet strategy, in which a Russian-backed national army would start a local conflict that would upset the established status quo. Damningly, in the Soviet zone, which had been reorganized and proclaimed the German Democratic Republic, a Reichswehr-style border police had been created. This police had military style weapons and training, but little equipment heavier than rifles and mortars. While not the equal of any Western military, when compared to what West Germany had (fuck all), this was considered a potent force. Specifically, Western planners feared that the Soviets would unleash this forces against the Federal Republic in a localized conflict which would threaten the "freedom" of the FRG, but would be small enough as to prevent a general war.

Underlying all this was one further fear, that of the Red Army. The Soviet forces in the East outnumbered the European-American forces in the West by well over 2:1, and at times closer to 4:1 in terms of divisions. In addition, the Soviets had divisions in Poland, Czechoslovakia, and Russia itself. Then add in the forces of the remilitarizing East Germany, and an unpleasant concentration of forces faced Western strategists. Through all this, there was only one real way to counter the Soviet threat, increase the number of full-strength divisions stationed in Europe. But when people looked around and asked who should provide those divisions, nobody (including the US) wanted to further increase their military spending. The only realistic solution was to get the Germans to provide men and weapons for their own defence.

All that is a very complicated way to provide context for a set of complicated negotiations in which the coalescing NATO powers had to ask Germany to rebuild its army. Essentially, the US and the UK wanted Germany to build a new army along the lines of the old Wehrmacht and Imperial Armies. France, who had been on the receiving end of those two armies in the last 50 years, instead wanted a multi-national force in which Germans provided the manpower for an inter-European army. And the FRG, for its part, wanted to get out of its "dog house" and become a real country again. Essentially, the Germans agreed to create a new army in exchange for an end to the occupation and a set of agreements which permitted it to become an equal partner in Europe. The US, UK, and the FRG all got what they wanted, and France got dumped on.

So to summarize everything, the FRG Constitution and the peace treaty after World War Two never limited the German Army because it was very much needed to participate in the Cold War.

Japan:

Ill admit that the Japanese peace treaty is a bit out of my wheelhouse, and I would love for somebody else to flesh out this response a bit. But very basically, most of the pressures I illustrated above didnt exist for Japan. Most importantly, Japan was seen advantageous not for its military but for its geographic position. Thus, rather than allow Japan to remilitarize against a common foe as Germany did, Japan's remilitarization was strongly managed. Most importantly, the Japanese constitution included Article 9:

ARTICLE 9. (1) Aspiring sincerely to an international peace based on justice and order, the Japanese people forever renounce war as a sovereign right of the nation and the threat or use of force as means of settling international disputes.

(2) In order to accomplish the aim of the preceding paragraph, land, sea, and air forces, as well as other war potential, will never be maintained. The right of belligerency of the state will not be recognized.

Essentially this clause has been subsequently interpreted as swearing off all forms of offensive war. Thus, when Japan remilitarized it created a "Self-Defense Force", an army whose only potential theater was the Home Islands. Likewise, the Japanese Navy and Air Force focused on defending Japanese territory, not supporting campaigns abroad. This is also the section which has recently been reinterpreted, to much controversy, in the last few years.

Also, I would like to point out that all my answers here are a sort of top-down view of these processes. Its important to keep in mind that the people of both countries had strong feelings concerning war and the use of militaries, especially after World War Two. Germany had been ripped apart and Japan had been nuked. Both countries had suffered millions of casualties and had been completely laid low by their defeats. Many people believed that these experiences taught a fundamental lesson: that war could bring nothing good. Especially in Germany, remilitarization was a challenge simply because people didnt want to repeat the mistakes of 1914 and 1939. They swore off war because many felt that it had brought their countries nothing but death and ruin.

Sources:

Konrad Adenauer, Adenauer Memoirs 1945 - 53

Dean Acheson, Present at the Creation: My Years in the State Department Robert McGeehan, The German rearmament question;: American diplomacy and European defense after World War II

A. J. Birtle, Rearming the Phoenix: U.S. Military Assistance to the Federal Republic of Germany

Kevin Ruane, The Rise and Fall of the European Defense Community: Anglo-American Relations

Michael Creswell, A Question of Balance, How France and the United States Created Cold War Europe