It would probably take hours to list the differences between WinRT and Win32, plus it would probably span hundreds of pages. Win32 is a very old API that has been pretty regularly updated to include just about anything. It really is quite a massive achievement and easily among the most powerful APIs for any system on the planet.
WinRT is a new API introduced with Windows 8 and it lacked maturity. As a result of this it did not have many features that are essential for making a powerful application. WinRT was designed for simple apps that utilised a mainly touch first user interface. Things have begun to change with Windows 10 and the WinRT has continued to be developed and improved. This improvements have put a much greater emphasis on the abilities of the API and it is now much more powerful but certainly nowhere near to the level of Win32. Don't be expecting Photopshop, 3DS Max or even simple apps like CCleaner to be making the transition to WinRT any time soon.
The APIs on iOS are obviously more mature and more feature rich because they have quite simply had more time and more use by developers. The same applies to Android. APIs themselves are not strictly necessary to make feature rich apps. It is possible to code many things manually but to do so would add months and in some cases even years onto the development of an application. APIs are incredibly useful in allowing the rapid creation of so many applications that are both powerful and efficient. This principle doesn't always apply to the modern environment though because all WinRT apps run in a sandbox so it is difficult to directly interface with the system and other apps. For example, the Adobe Application Manager as a modern app would not work because it wouldn't be able to access any files relating to other Adobe applications because they would be outside of the sandbox.
I am not the most knowledgeable about this aspect so hopefully someone with much more experience will be of better assistance to you.
Regards,
Aaron Graham.