It's pretty easy to keep a list of old salted hashes, and copmare the new password's salted hash against the previous ones. It doesn't require saving the old passwords.
To compare your new password with your old password, you take the old salt, hash your new password together with it, and compare the result to the old hash. If they match, you're trying to reuse the same password. You do this on the server side, naturally.
If salted hashing were done on the client side, it means you're actually sending username + saltedhash, instead of username + password to the server to log in.
So an attacker could submit a precomputed or stolen salted hash to be compared against the stored one -- completely defeating the point of hashing passwords in the first place.
>So an attacker could submit a precomputed or stolen salted hash to be compared against the stored one -- completely defeating the point of hashing passwords in the first place.
You could hash once on the client and once on the server to get the best (?) of both worlds. Really only the server one needs to be salted.