The question is asking whether full-time indoor cats are as happy as indoor/outdoor cats. The reason why the question has been asked is because people make the presumption that if a cat is allowed to express their natural behaviour they’re more likely to be happy. And it hints at the argument that a cat allowed outside has a better chance of expressing their natural behaviour than a cat inside the home full-time. However, the answer to the question is not straightforward because you have to factor in health and welfare as well as happiness.
In general, you have to agree that it is more likely that a cat allowed outside is going to be happier if happiness is equated with behaving naturally. But this conclusion must be modified by at least a couple of factors.
Firstly, a lot of full-time indoor cats live in homes where their human caregiver spends a lot of time with them, entertains them and where the home has been modified to accommodate natural cat behaviour. The cats are stimulated and therefore it is no great loss to them that they cannot go outside.
Secondly, an injured or sick cat is going to be unhappy. There is a greater possibility that a cat allowed outside is going to become sick or injured. This is because they can encounter other cats and those cats might be ill with a contagious disease. The cat in question may contract that disease and become seriously ill. They will be unhappy. They will be less happy than a full-time indoor cat who is healthy but possibly slightly bored. In this scenario the full-time indoor cat is happier than the outdoor cat.
Sadly, domestic cats allowed outside get hit by cars. It must be the single biggest danger to outside cats other than the coyote in America which preys upon domestic cats and other animals. A dead domestic cat cannot be happy. The cat being chased and attacked by a coyote cannot be happy. Therefore under this scenario the full-time indoor cat looking out a window, longingly expressing a desire to be outside (it seems) is going to be happier than the poor cat who’s been attacked by a coyote.
You can see, therefore, that the answer to the question is not straightforward. On a straight like-for-like basis in which (1) both the full-time indoor cat and the indoor/outdoor cat are healthy and safe and (2) the indoor cat’s owner makes no real effort to entertain their cat then the indoor/outdoor cat is going to be happier.
If the question is asking whether a cat living outside all the time is happier than a cat living inside then you have to say no. This is because a cat living outside full-time is going to be a stray cat and it is likely that they will be struggling to survive in terms of finding food and keeping warm and dry. That said, there are some individual domestic cats who voluntarily leave their cosy, warm home where they are well cared for to live in the wild. This is unusual but it does happen. These cats are happier living in the wild.
Postscript: as an afterthought, I have to ask whether a domestic cat is less happy if their life is tougher and more difficult than it would be if well cared for by a person. It is not automatically the case that a cat will be unhappy if life is tough. We have to be careful not to project human thoughts onto our cats.