If two passwords match, e.g. because a user has multiple accounts and use the same password for all, this can easily be spotted if a constant "salt" is used.
Another way of looking at it is that a hash function that is applied to passwords with a constant salt, is equivalent to using a modified hash function without salt. In the case of the MD and SHA family of hash functions, you get such a modified hash function by modifying the initial state, round constants, round functions and message expansion.