Just on a practical level... Pavarotti could not sing like Justin Bieber. Pavarotti was just generally not very good at singing contemporary music. Great opera singer? Yes!!!! One of the greatest!!
The skill of 'singing' just on a real world, common sense level does not translate. Otherwise you would be seeing hundreds and hundreds of professional cross over artists. But you don't. You see either a singer who is a master at doing ONE thing or you see a singer who is good at doing many.
They're different skillsets and depending on your goal, you might want many different skill sets. You might make a career out of doing both; that's what makes you unique and sets you apart. But I just don't hear the operatic tenor performing at the MET every night who, the following night, is singing pop music at a sold out venue. It just doesn't happen.
Now if you want to get specific and ask about the physiological differences, there are a plethora! From supglottic pressure, to vocal fold closure, to formant tuning strategies, breathing strategies, lung volume differences, vibratory mechanism differences, etc. Plus the way our body responds in terms of muscle development and muscle memory. The list goes on so long that you could write books on the differences and still we wouldn't be able to completely grasp the complexity of it.
Long distance running and sprinting are basically the same sport. Whoever can run to the finish line the fastest wins. But the way the body of a distance runner responds to distance running is massively different from the way the body responds to sprinting. They recruit different sets of muscle fibers. The stress on the body is also radically different as is it's hormonal response. Not to even mention requiring mastery of two completely different sets of techniques.
Just because you are a good distance runner does not mean you will be a good sprinter How many Olympic level athletes are doing both?
Is Usain Bolt spending his time working on his distance running thinking it will improve his sprinting? Or is he sprinting to improve his sprinting? Does Usain Bolt have the mindset of a sprinter or a guy who dabbles in some of everything?
The only reason people study classical is because they have this impression that classical singers are somehow superior to contemporary singers. That if you learn to sing opera it will help you sing Taylor Swift because opera is an elite form of singing and Taylor Swift is not. And, you know what, go for it! If you believe that then go for it.
Me, personally, I'm not interested in being that guy who is a decent sprinter and a decent distance runner. I want to be really really good at sprinting and say fuck distance running. I want to emulate what successful sprinters are doing, not re-write the book on sprinting to include distance running. That's not my thing.
If other people want to train both, great! No worries! So long as you're doing it out of choice not because someone told you to.