None.
Public and private key are used to negotiate a symmetric encryption key, that will be used during that session. Asymmetric encryption uses two keys (private and public), symmetric encryption uses only one.
Your server sends its public key to a client, the client validates that key signature (check the CA and all that) then use it to encrypt a randomly selected key that will be used as symmetric encryption key, and send it to the server. Because only the private key can decrypt that message, the message is secure, only the server can decrypt it. Then the server accepts that key selected by the client, and they start to transmit data using symmetric encryption.
Why all this? asymmetric encryption is quite computionally expensive, so it is just used to ensure that client and server can negotiate a secure symmetric key without sending it in plain text. Symmetric encryption is cheap.
Symmetric encryption is also safe, the problem is that both parts must know the key before start, and that is a big issue. By using asymmetric encryption for negotiating the key, this problem is solved.
UPDATE
Well, it seems that @EJP does not agree with my answer, so I tried to find some more documentation that explain the thing in an easy way.
http://www.techradar.com/news/software/how-ssl-and-tls-works-1047412
SSL explained
When you visit a bank's website, the bank's server will automatically
redirect you to its secure site using the HTTPS protocol before you
can log in. This results in your browser and the bank's site
negotiating a secure channel using SSL.
This negotiation goes a little like this (note that I've simplified it
greatly). The browser sends a message stating what the latest version
of SSL it can support and a list of symmetric algorithms it can use.
The web server sends back a message with the version of SSL and the
algorithm that will be used.
It sends its certificate as well. The client verifies the certificate
using the known certificates that came with the browser; in other
words, it checks that it has been signed by a trusted CA and that it
hasn't expired.
If the certificate is valid, the browser generates a one-time key for
the session, encrypts it with the server's public key (it's part of
the certificate), and sends it to the server. The server decrypts the
key, then uses that key together with the agreed symmetric algorithm
for the rest of the session.
I may be confused.