This question concerns the session send and receive keys used in SSL/TLS protocol. my understanding is that this key uses symmetric encryption (DES, AES, BlowFish, etc.) I'm wondering, if public-private key pairs are superior to symmetric keys regarding key exchange security, why don't use asymmetric encryption for the session keys too?

this is an extension of an existing question: security of PKI, Certificates, certificate authorities, forward secrecy

3 reasons (now):   


Asymmetric encryption is slower, much slower, than symmetric encryption. Orders of magnitude slower. 
Given the same keylength, asymmetric is much weaker than symmetric, bit-for-bit. Therefore, you need a much larger key to provide equivalent protection. This also contributes to the slowness mentioned in 1.
(As per @ThomasPornin's comment:) Asymmetric encryption carries with it an increase in size of output. For instance, if you use RSA, encrypted data is at least 10% larger than the cleartext. Symmetric encryption, on the other hand, has a fixed size overhead even when encrypting gigabytes of data.

