I don’t think one is objectively better than the other. Both are amazing zoos, and while each contains a few outdated animal exhibits, the majority at both are modernistic and beautiful.
However, if I were to be asked which I personally prefer, I would say San Diego as it’s a larger and I feel more “complete” collection. For instance, Bronx has only one bear, four cats, and one great ape, whereas San Diego has five (until recently six) bears, at least 13 cat species by my calculation, and three great apes. Bronx has no hippos while San Diego has both species. Which isn’t to say San Diego outclasses Bronx in every category; for example, the latter has two rhino species while the former currently has none; if I’m not mistaken there are no large ratites at San Diego while Bronx has ostriches and emus; Bronx has two penguin species but San Diego only has one; San Diego only has one large canid while Bronx has two, etc. And then in certain areas, like non-ape primates and crocodilians, both parks have impressive collections. But overall, I would say San Diego caters more to my personal tastes.
It should also be noted that this may not be the most fair comparison considering how San Diego Zoo’s climate allows it to keep many more warm-climate animals outside that would have to spend a good portion of the year indoors in New York. Nonetheless, I stand by personally enjoying San Diego more.