I'm on mobile and don't remember the source exactly, but German paratroopers were dropped with an Irving style harness that attached at one place on their back, requiring them to take a long time detaching it and forcing them to land on their hands and knees.
Allied paratroops used a parachute harness that attached at the shoulders, meaning they could drop with their hands free and release their parachute immediately upon landing.
The different styles of parachute meant that German paratroops were dropped relatively unarmed (holding just a knife or a holstered pistol) with the rest of their equipment dropped along with them in containers that they had to find and reopen, whereas allied paratroopers could hold their own weapons as they came down. This made allied paratroopers far less vulnerable when dropping into contested territory, as they tended to do.
You might be better off asking about more specific differences. There are many intrinsic differences that just come with the fact that they fought for different nations. They have the obvious differences like weapons, aircraft type, and to some extent their organization but they also have less obvious differences like tactics, or jump routines/styles. Asking a more specific question may also allow for people knowledgeable about just one of the three organizations to answer. For example I'm knowledgeable in American paratroopers but know diddly-squat about the other two organizations. I could answer the specific question as it pertains to the Americans and someone else could answer as it pertains to one of the other nations. I'm sorry this isn't an answer but I hope it helps.
A quick look into the organization of the German and British Parachute Battalion shows that the German unit was huge in comparison to it's Allied counterparts. The German Battalion had 850+ men in comparison to the Allied Battalions which both came in around 600 men. This meant that a German Fallschirmjager Regiment had almost 2600 men while the British had just over 1800 and the Americans had about 1700 men (not counting regimental level support units in any cases).
Main difference seems to come from the fact that the Germans had more non rifle company elements and more men in these elements. For example. The Battalion HQ for the Germans had a HQ section (5 O, 20 men), a Communications Platoon (1 O, 41 men), and a Battalion Supply Train (2 Officers, 69 men) for a total of 8 O and 130 men. The Brits had the HQ section with 5 O and 22 men, and then in a separate HQ Company they had an Intelligence Section (1 O, 8 men), a Signals Platoon (1 O, 27 men), and an Administrative Platoon (3 O, 62 men), for a grand total of 10 O, and 119 men. Us Americans had just 6 O for some reason in Battalion HQ plus a Staff Section (15 men), a Mess Section (22 men), a Communication Section (27 men), and a Supply Section (13 men) in the HQ Company for a total of 6 O and 77 men.
Here is a cool source that you can check out for some comparisons: US
Keep in mind though that these are the regulation numbers and not the actual size. The American version I linked is the version that was official until the end of 1944. In reality though units in the field were stylized more on the version that became official in after 1944. They gained over 120 to go from 580 men to 700 men per regulation but the regulation was only to make the numbers/arrangements in the field "legal".