I think that terms are being confused here everywhere. You guys are referring to two totally different things.
1- You are asking about difference between 'focus ring' and Aperture priority.
2-sant kabir is referring to OLD lenses that had also an aperture ring on the lens (besides the focusing ring)
Now. Focus ring has absolutely nothing to do with Aperture priority and, in general, you can not use one to substitute the other.
Focus ring simply "decides" where the focus is at any time. The aperture used decides "how much" is in focus in front and behind of the focus point.
The only time that focusing sort of resembles the functions of the aperture is when you want to have a large depth of field and want to use a small f number (large aperture) BUT you have to take many photos, stack them all together and blend them without ever touching the focusing ring but only but moving the camera in the proper axis (Very used in macro photography). If you don't do it that way, you won't be able to properly blend the images.
If you have more questions or still don't understand very well, let me know and I will give you a hand.
Again, to summarize:
Aperture decides depth of field.
Focus ring decides where the focus is set (no matter what the depth of field)
Edit #1: Ok, seeing your example. It seems there is still a bit of confusion but we are getting there.
Let's use the example of your flower. Flower is there, you get pretty close to the flower and you use the smallest f-stop number (that is the biggest aperture of your lens) then you use the focus ring to focus on a certain spot of the flower. It is most likely that except for what you are focusing on, the rest of the flower will be out of focus and blurred. For a close-up photo of a flower where you want everything out of focus, except for the flower itself, you either use the method I mentioned earlier or get the flower very well separated from the background and chose a proper focal distance/aperture combination to guarantee that the flower and nothing but the flower is in focus.
The reason why you can not use them interchangeably is: Say your aperture is on X and your focus is on distance Y but there is nothing at that distance, then everything is out of focus. Now, let's say that something is at that distance Y, then whatever is that that distance is in focus a certain amount in front of and behind Y and that will be given by the X you selected.
In other words. If you don't focus, you will never accomplish focus using Aperture priority. (This is a bold statement since having a wide angle lens on a large f-number will guarantee that almost everything is in focus in your photos).
The nature of Aperture and Focusing is optically different. And that's why you cannot use them to substitute the other but to complement each other. Aperture uses the diffraction/or not of light while focusing refers to ray of lights that actually converge
Pfff... I feel like I wrote an essay here! Again, if you have more questions... you know...
Check the extra link I added... It will show you examples.
For more info, check these links, they will help you understand: