[Kelly Crooks]

One of the differences between symmetric cryptography uses the same key to encrypt and decrypt data. Asymmetric cryptography uses a pair of keys, one public and one private to decrypt the data
Another difference is that symmetric cryptography uses complex and faster algorithms than asymmetric. Asymmetric cryptography uses a more complex and much slower algorithm. As the number of users grows with symmetric cryptography so does the number of keys. Asymmetric does not grow uncontrollably and has one pair of keys per user. Symmetric cryptography key exchange is out-of-bound and possibly insecure. Asymmetric cryptography key exchange is a public key that is safely distributed widely.

Symmetric Cryptography pros:
Fast and convenient to use
the method is easy and simple to understand

Symmetric Cryptography cons:
The receiver must get the secret key from the sender

Asymmetric Cryptography pros:
Easily Scaled
Only one pair of keys for each user
Does not require out-of-band delivery of key
Provides added functionality for commercial use
Authentication
Access control
Confidentiality
Integrity
Non-repudiation

Asymmetric Cryptography cons:
Slower than Symmetric by up to 1000 times
Requires validation of key pair ownership
Exposure, destruction, or loss of private keys destroys the integrity of the system