Why were the Germans during the Nazi era so inventive?

The Germans weren't broadly more inventive than the British or the Americans. What they did have is different national priorities and desperation for wonder weapons to win the war which lead them down paths the Allies didn't bother with. Couple this with madmen in control of national budgets and they'd fund all sorts of crackpot ideas. Few worked out. Mostly they were a waste of war resources.

We tend to fixate on the fastest and biggest weapons: the biggest tanks, the fastest jets and rockets. But wars aren't won by wonder weapons, they're won by efficient weapons. It doesn't matter who invented it first, or who had a flashy prototype like so many German inventions during the war. The weapon has to work and you have to be able to produce it without bankrupting the country. The Greeks invented the steam engine, but it took 1700 years for it to be made practical. The Germans had all sorts of ideas about space, but it took 20-30 years and large chunks of the budgets of two superpowers to make just some of them happen and the rest are still on the table. The really hard part of invention is making it reliable and economical. It takes a lot of time, a lot of money, and a lot of people.

The Allies were well ahead of the Germans in computers, signal intelligence, radar, electronic miniaturization and nuclear weaponry. The Germans were well ahead in rocketry and in jets by about a year because they were a national priority. Let's look at them in some detail.

German liquid rocketry was fueled by Hitler's desire to have a vengeance weapon to strike at London and other strategic targets. The Germans lacked a heavy four engine bomber and, after 1941, the air superiority to deploy it. They were losing the air war, so they needed to try something else. Rocketry was heavily funded to produce the V2 rocket. Militarily it was useless, being wildly inaccurate. Politically it failed in its mission to bring the UK to its knees. It was a colossal waste of resources for Germany, but the rest of the world benefited by using the V2 design, and German rocket scientists, to get a jump start on their space programs.

The Allies had invested in heavy bombers and so had no need for a strategic rocket. They instead developed small rockets for ground attack.

German jets and solid rocketry were fueled by a desperate need to stop the around the clock Allied bombing campaign. Germany need a cheap, high performance interceptor that could reach the high flying and heavily armed Allied bombers. Range was not important, they were flying over Germany, so fuel hungry jet engines could be used. Many, many designs were tried. Only two were successful: the Me 262 and (to a much lesser extent) the He 162, but they came too late to change the war.

In contrast, the Allies needed long range fighters to protect their bombers; fuel hungry jet engines would not make it to Germany and back. Instead, they opted to stick with conventional, fuel efficient propeller driven aircraft. By the time the Me 262 appeared in 1944, there were so few of them with so little fuel and so few experienced pilots, slower Allied fighters overwhelmed them. The Allies quickly realized the jets had short legs and learned to follow them back to their bases and shoot them down as they landed.

But the Germans were only about a year ahead in jets. Most Allied nations were developing jets during the war and were operational shortly after. The Gloster Meteor was operational in 1944 and the P-80 Shooting Star was prototypes. With a range of just 600 miles, compared to the P-51 Mustang's 1500, the Meteor was relegated to defending the UK.

Synthetic fuel wasn't new, but the Germans greatly advanced the art of producing it driven by their extreme lack of oil. The US had their own program but with plenty of oil it remained lackluster.

Instead, the Allies focused on synthetic rubber. Japan's conquests had cut the Allies off from the world's supply of natural rubber and they needed a way to make tires for all those vehicles and gaskets for all those engines. By the end of the war, synthetic rubber production was twice the natural production before the war.

German guided weapons were driven by their desperate need to stop the overwhelming production of Allied armor, ships and bombers. Infantry and pilots needed a weapon that could reliably destroy them outside the range of their defenses. Thus an array of guided weapons were trialed by the Germans. Many used terrifyingly unstable fuels. Most did not see operational use and were cancelled with only the anti-ship missiles Fritz X and the Hs 293 seeing significant deployment.

The Allies were doing a fine job destroying German aircraft and tanks, but their bombers were woefully inaccurate. Their guided weapons focused on precision bombing. The Azon and the Bat were operational during the war. The GB-8 and VB-6 Felix were beginning production as the war ended.

In contrast, the British (and then the Americans) were well ahead of everyone else in radar. Radar had been around decades before WWII and the Germans had their own radar installations but it wasn't very good. It took the British to perfect it. They made radar a national priority to defend their island nation from air attack as well as to give them an edge in naval combat. The Americans were a naval power and wanted radar to detect ships and aircraft as well as to direct artillery fire. Collaboration between the two countries advanced radar by leaps and bounds.

By the end of the war, Allied radar and electronic miniaturization was so good they could pack it in anti-aircraft shells to explode when near an airplane and in artillery shells to burst above the ground.

While all nations developed radar guided night fighters, it was the Germans, with their increasingly desperate need to shoot down British bombers operating at night, who produced the most night fighters. However, they were all refits of existing heavy fighters such as the Do-217J. The ultimate expression of the night fighter in WWII was the American P-61 Black Widow.

Similarly, the pressures of the Battle of the Atlantic drove the British to invent and develop the first computers to speed the breaking of German encryption. First the Bombes and then Colossus the first programmable digital computer.