There's one, admittedly theoretical difference, at least: there is such a thing as perfectly (information theoretically) secure symmetric encryption scheme (regardless of the computational power of the adversary), but asymmetric crypto, by definition, relies on being computationally infeasible, not information theoretically impossible.
Even though it's theoretical, the side effects of this fact surface from time to time as engineering issues in asymmetric crypto: all information that the attacker might need to break asymmetric crypto is more or less in the ciphertext, intuitively suggesting it's easier for asymmetric crypto to catastrophically go wrong.
No, I still don't get it. How is information about the private key in the ciphertext?
It's great that one-time pad exists, but it's not really relevant in actual crypto code, right?
The only actual reason I can think of is that symmetric crypto is easier to write and understand - you just mangle and xor some text back and forth, while in asymmetric crypto, you need to understand fairly complex algebra. But again, that's not that important if you use existing primitives, right?
Even though it's theoretical, the side effects of this fact surface from time to time as engineering issues in asymmetric crypto: all information that the attacker might need to break asymmetric crypto is more or less in the ciphertext, intuitively suggesting it's easier for asymmetric crypto to catastrophically go wrong.