Walk behind mower or riding mower?
Walk behind mowers with engines low to the ground benefit from oiled foam more so than a riding mower.
My preference is oiled foam for walk behind machines and whatever for riding mowers although, I'd lean toward oiled foam there too. The main reason is that as machines age, paper element filters get harder to find. Not impossible, just harder. Whereas a foam filter, if properly maintained and not abused will usually last the life of the machine.
Which leads us to another aspect - maintenance. If you don't like to get your hands dirty and don't mind shelling out ten to fifteen dollars per filter, go for paper element filters.
A tip for those who prefer oiled filters. You can reduce the amount of exposure to oil if you put the cleaned filter in a zip lock plastic bag, add the oil, close the bag and distribute the oil into the foam by squeezing the bag. Don't add too much oil. A teaspoonful for a small filter and maybe a tablespoon for larger filters. I used to keep a spare foam filter ready to go in a plastic bag so that I could soak the dirty filter in varsol for a few minutes and clean it at my leisure.