KDC is suitable for smaller infrastructures where you place explicit trust into each person or node doing encryption. Each time Alice wants to encrypt a message to Bob, she has to ask KDC for a temporary key to use for encryption. The KDC will also need to provide that temporary key to Bob so he can decrypt the message. You don't want to just give Alice Bob's encryption key, as then she would be able to read all encrypted messages sent to Bob (the nature of symmetric encryption is such that the same key both encrypts and decrypts the message).
As you can imagine, this does not scale beyond local infrastructures, nor is very fault-tolerant. If the Internet relied on KDCs in order to do encryption, it would be easy for attackers to DoS them and kill all commerce on the web -- or at least make it unreliable enough not to bother. This is why browsers went with Public Key cryptography and a framework of mutually trusted certificate authorities (CAs). This has its own set of trade-offs. One, you have to trust that the CAs know what they are doing -- a trust they have repeatedly violated. Two, PKI relies on the hope that we'll never find a fast way to factor the product of two very large prime numbers -- a problem not present in symmetric cryptography. If the likes of Grigory Perelman one day find a way to quickly solve that problem, anyone will be able to obtain the private decryption key from the public encryption key. If that happens, we'll be all in big trouble and will have to come up with some other way for two untrusted parties to exchange encryption keys.
So, to answer your question -- cryptographically, KDCs and symmetric keys are stronger than asymmetric public-private keys, as there is no danger that one day some crazy mathematician will find a way to quickly factor products of large primes. On the other hand, KDCs have inherent problems with key distribution, reliability and ongoing trust that can't be easily solved and therefore KDCs are not suitable beyond local installations where such trust is easy to assure.