Symmetric cryptography ciphers are generally fast and hard to break. So speed is one of the key advantage of Symmetric ciphers and NOT a disadvantage. Symmetric Ciphers uses simple encryption steps such as XOR, substitution, permutation, shifting columns, shifting rows, etc… Such steps does not required a large amount of processing power compare to the complex mathematical problem used within Asymmetric Ciphers.
Some of the weaknesses of Symmetric Ciphers are: The lack of automated key distribution. Usually an Asymmetric cipher would be use to protect the symmetric key if it needs to be communicated to another entity securely over a public network. In the good old day this was done manually where it was distributed using the Floppy Net sometimes called the Sneaker Net (you run to someone’s office to give them the key).
As far as the total number of keys are required to communicate securely between a large group of users, it does not scale very well. 10 users would require 45 keys for them to communicate securely with each other. If you have 1000 users then you would need almost half a million key to communicate secure. On Asymmetric ciphers there is only 2000 keys required for 1000 users. The formula to calculate the total number of keys required for a group of users who wishes to communicate securely with each others using Symmetric encryption is Total Number of Users (N) * Total Number of users minus one Divided by 2 or N (N-1)/2
Symmetric Ciphers are limited when it comes to security services, they cannot provide all of the security services provided by Asymmetric ciphers. Symmetric ciphers provides mostly confidentiality but can also provide integrity and authentication if a Message Authentication Code (MAC) is used and could also provide user authentication if Kerberos is used for example. Symmetric Ciphers cannot provide Digital Signature and Non-Repudiation.
Reference used for theis question: WALLHOFF, John, CBK#5 Cryptography (CISSP Study Guide), April 2002 (page 2).