What specifically is the advantage of the version with calories?

Well, let me consider a recent, highly upvoted post on here: A Contamination Theory of the Obesity Epidemic. In it, the author says that the explanation for the obesity crisis can't be CICO,

“It's from overeating!”, they cry. But controlled overfeeding studies (from the 1970's—pre-explosion) struggle to make people gain weight and they loose it quickly once the overfeeding stops. (Which is evidence against a hysteresis theory.)

“It's lack of exercise”, they yell. But making people exercise doesn't seem to produce significant weight loss, and obesity is still spreading despite lots of money and effort being put into exercise.

If CICO is literally true, in the same way that the "atoms in, atoms out" theory is true, then this debunking is very weak. The obesity epidemic must be due to either overeating or lack of exercise, or both.

The real debate is, of course, over which environmental factors caused us to eat more, or exercise less. But if you don't even recognize that the cause must act through this mechanism, then you're not going to get very far in your explanation. That's how you end up proposing that it must be some hidden environmental factor, as this post does, rather than more relevant things related to the modern diet.

My own view is that the most likely cause of our current crisis is that modern folk have access to more and a greater variety of addicting processed food, so we end up consistently overeating. I don't think this theory is obviously correct, and of course it could be wrong. However, in light of the true mechanism behind obesity, it makes a lot more sense to me than many other theories that people have proposed, especially any that deny we're overeating from the outset.