This is a very tricky question and I’m not completely sure of the answer but there are at least two pieces of information which indicate that cats do indeed have less health problems than dogs. Note: it is disputable that cats are healthier than dogs so this is a discussion article, no more.
The first is that on the AVMA (American Veterinary Medical Association) website they say that during 2011 44.9% of cat owners did not take their cat or cats to a veterinarian whereas 18.7% of dog owners did not take their dog or dogs to a veterinarian during the same year. Therefore a lot more cat owners did not take their cat to the veterinarian than dog owners which implies that dogs are less healthy than cats because the main reason that people take their cat or dog to a vet is because the animal is sick or injured.
The second main reason, as far as I can tell, why cats might be more healthy than dogs is because cats are more resistant to diseases carried by arthropods which are insects such as mosquitoes, fleas, ticks, lice and flies. A study dated 2016 concluded in the abstract that:
“Cats have greater genetic diversity than dogs with much lower linkage disequilibrium in feline compared with canine breed groups.”Study: Cats are not small dogs: is there an immunological explanation for why cats are less affected by arthropod-borne disease than dogs?
They say that although the immune systems of cats and dogs are similar there are differences in the immune function of the animals which they think accounts at least in part for the differences in autoimmune, allergic, idiopathic inflammation, infectious and immunodeficiency disease occurrences in these species .
Dogs have been selectively bred for thousands of years longer than cats. There are far more dog breeds than cat breeds. Selective breeding can weaken immune systems due to inbreeding depression. I am speculating here but it is conceivable that cats are more robust that dogs generally because in developed countries (and this comparison concerns developed countries) the proportion of random bred cats in the entire population is much higher than for dogs. This means greater genetic diversity in cats which in turn means a more robust immune system and better health.
P.S. I’d appreciate the input of others in FB comments 😉