What is the difference between PBE and symmetric key encryption? What is the difference between PBE (password based encryption) and symmetric-key algorithms? 

My understanding is that symmetric-key algorithms like AES and Twofish are used in PBE. Hence PBE is just a technique which employs symmetric-key algorithms to do the work. So there is no point in comparing the two. Is that correct, or am I missing something?
As typically implemented, PBE takes a low-entropy, user-supplied password, adds some entropy to it, and thus strengthens it before turning it into a key. This key can then be used for symmetric encryption.

The problem is that the user's password often has so little entropy to start with. If an attacker learns the salt, digest method and quantity of iterations, he needs to keep trying to guess the pre-strengthened password, apply matching PBE-strengthening to his guess, and then use the resulting key to match up with ciphertext that was created with that key.

If the programmer used a decent digest method (such as PBKDF2 with HMAC SHA 256), a salt generated by a cryptographically secure pseudo-random number generator (CSPRNG) and a decent quantity of iterations (200,000 or more) then he can make it pretty expensive for attackers to guess the key.

However if the user's password is ridiculously easy to guess (such as "password") then even a good digest method and lots of iterations won't make much difference. You still need to force the user to choose sufficiently complex passwords, to maximize the entropy prior to any strengthening occurring.

OWASP has some best practices on enforcing password complexity.