... when the client could just create a random symmetric key and encrypt it using the servers asymmetric public key?
What you are describing is essentially RSA key exchange which is also available, at least until TLS 1.2. The problem is that this kind of key exchange provides no forward secrecy, i.e. if the attacker manages to get the private key of the server he can extract the symmetric key from the key exchange to this server and can thus decrypt any previously sniffed traffic to this server. Contrary to this DH results in a key which is not associated with the servers key or any other static secrets and thus provides forward secrecy.
Even if the attacker has access to the private key and only wants to sniff the current connection DH makes it harder. With DH key exchange the attacker needs to be in the middle, i.e. must create a TLS connection between client and attacker and another one between attacker and server, both using the same certificate and private key so that the victim does not notice. Depending on the position of the attacker in the network this active attack can be easy or not. Also, active MITM is not possible if the client is also using a certificate (mutual authentication) unless the attacker also has access to this certificate. With RSA instead the attacker only needs to passively listen to the direct connection between client and server in order to decrypt it and can do this also when client certificates are involved.