If you are running the same version, you shouldn't have to do anything, they should be the same.
More importantly you should NOT be using PASSWORD() to encrypt user passwords. The algorithm changes from mysql version to version. MySQL specifies in the manual it should not be used for the purposes you are using it, but only creating user privileges to mysql itself.
Also note it is not as strong encryption as SHA1 which is most likely the algorithm you should be using to hash your passwords.