The question in the title implies that wild cats have round pupils. This is incorrect. Although if we look at photographs of the pupils of wild cats there is no doubt that the pupils of lions are much more circular than the pupils of pumas which in turn are more circular than the pupils of domestic cats which are positively elliptical with sharp ends.
The point I’m making is that there is no fundamental difference in terms of eye anatomy and mechanics between wild cats and domestic cats. I have to conclude that even the lion and tiger have elliptical pupils and not round pupils but you have to look carefully at the photographs to see that the pupils of lions are not precisely round.
The best-known book on the wild cats by Mel Sundquist and Fiona Sundquist does not distinguish between the anatomy of the eye of wild cats and anatomy of the eye of the domestic cat. They say with respect to wild cats:
“On a sunny day, a cat’s pupils contract to a vertical slit or pinprick, while in dim light the pupils dilate a large circular or oval…”
I have to say by the way that I do not like the reference to “pinprick”. I believe this is contradictory.
Why are the pupils of lions more round than those of the domestic cat? I would argue that, in very bright light, the domestic cat is able to shut out more light from the retina than the lion. The domestic cat can do this by retracting their pupils to a very narrow slit and then lowering their eyelid over the slit to further regulate and limit the input of light which impinges upon the retina. As the pupil of the lion is rounder, this wild cat is less able to limit (at the extreme) the amount of light into the eye.
On this basis, I would argue that the retina of the lion is possibly slightly less sensitive to that of the domestic cat. If this is true I am not sure why. The lion hunts at night and in the daytime (but is described as nocturnal) like the domestic cat. Therefore the sensitive of their eyesight should be similar. It may be to do with the high ambient light of the environment in which the lion lives.
The variations amongst the wild cats of the roundness of their pupils might be partly dependent on whether they primarily or exclusively hunt at night or during the day.
Update Nov 2017: Recent research indicates that the vertical slit provides the best view for stereoscopic vision and better depth perception which is important for small ground hunters. Also the wild cat ancestor of the domestic cat, the N. African wildcat hunts exclusively at night while the lion hunts primarily at night. This may indicate that the domestic cat has more sensitive eyes than the lion and tiger and therefore needs vertical slits to better protect the eyes in bright daylight as explained above.