How does an RTOS do it quicker than an interrupt?
It doesn't, an RTOS is no such thing, and will always be considerably slower than a properly written non-RTOS application on the same hardware.
Essentially (in my opinion) an RTOS is a 'sledge hammer approach' to a problem, throw massive hardware at it (600MHz ARM for example) and it doesn't matter much how sloppy your programming is
For those of you who remember the Commodore Amiga?, the OS was based on UNIX style commands, and much of the disk control system was actually stored on it's floppy disk (disk copy, file transfer etc.) - it was all written using Lattice C, and a bunch of enthusiasts rewrote all the disc based commands in assembler. This reduced the file sizes to about 25%, and increased their speed four or five times.
The same applies to Windows, Windows programming is incredibly sloppy, with absolutely massive file sizes and requiring incredibly fast and extensive hardware to run at a decent speed. Going back to the afore mentioned Amiga, it ran under 8MHz, only had 512K of RAM, yet produced amazingly fast graphics and games, because it was written in assembler and for efficiency.
I can remember the first CD-ROM's coming out - what a revelation, you could have all of your games and programs on a single disc - not for long you couldn't, more space simply means sloppy programmers use more of it, and it wasn't long before programs came on multiple CD-ROM's.
I wonder how fast correctly written DOS software would run on modern PC hardware?, with the huge overheads of Windows or Linux removed it would be pretty damn fast

For that matter, how fast would the above mentioned 600MHz ARM be without the crippling OS?.