Asymmetric Cryptography was the start of a new era in cryptography.
Diffie-Hellman key exchange is asymmetric cryptography.
We also use digital signature algorithms. These allow someone with a private key to digitally "sign" a message, such that anyone with the corresponding public key can verify that the signature was made on that message by that private key. This allows proving the authorship of messages and provides the non-repudiation. The non-repudiation is only possible with asymmetric cryptography and Rabin Signature Scheme was the first genuine digital signature.
Key exchange and digital signature schemes are the most common use of asymmetric cryptography.