AES in ECB mode weakness
Given only what you've said, and assuming the keys are created and stored in a strong manner, using a different key to encrypt database entries mitigates the problem of ECB mode. Namely that identical plaintext, when encrypted with the same key, always outputs the same ciphertext. No security is gained by switching to CBC mode (assuming you can easily store all the keys securely, but see what @Thomas Pornin had to say about that). The practical gain by switching to CBC mode is that you only have to store one key securely. The IV's don't typically need to be protected.

In the second scenario where the exact same key is used for all entries in ECB mode, the advantage the attacker gains is that if he knows a plaintext/ciphertext pair, he now knows everywhere that plaintext appears in the entire database. 

For example, lets say the attacker's own info happens to be in the database. He can look up the encrypted version of his gender. He now knows the gender of everyone else in the database (if the ciphertext is the same as his, he knows the entry is for a male, otherwise it is for a female). This same idea can be extended to other fields (age, first name, last name, etc). The key to this advantage is though that the attacker must have plaintext/ciphertext pairs.