FIrst off. I want to point out the UFC DOES NOT. DOES NOT have a patent on the octagon cage. THEY DO HAVE A PATENT on calling the said cage "THE OCTAGON".
The reason for this was, the Ufc wanted the uniformity to help the emerging sport earn more official sanctioning. Thats why in America, all mma orginzations that have a cage, have a cage that is in an octagonal shape.
Now to answer the question at hand more directly:
Ring fights and cage fights are different. Equally different.
They each have pros and cons.
The cage is held in high regard (especially by americans, i personnaly like it), because the cage itself can be utilzed effectivly. It helps box a person in a limits their escape....making it a great place for ground and pound.
The cage favors the agressor.
The ring makes it easy for someone to rest on. Much harder to takedown. Helps against ground and pund (u get on other side of ropes- aggresor has to punch through that).
the ring favors the defender.
People like the cage because it opens up vrand new possibilities in the fight and makes combat more resoureful.
People (americans mostly) like that alot.
People in the ring limit the possibilies but force the fighter to try to fight in the open space on the mat more. They want the fights in the center on the open mat. They dont want fight to go the edge or wall. They want to imagine a very neutral battleground. No advantages.
Thats why Pride (japanese people) favors the ring. They think its more fair.
I hope this helps.