SSL uses both asymmetric cryptography and symmetric cryptography.
Why can't it, or why doesn't it, just use one of them?
Symmetric cryptography cannot be used for exchanging secrets between machines that had never communicated before, but asymmetric cryptography is limited to encrypting very small data (the size of the key) and is extremely slow when used to encrypt larger blocks. That's why all practical cryptography scheme uses both.
Just to underline how TLS use this
Let's say we want just one of them: