Server does not accept public key for ssh login without password

Did you make sure that the ownership and mode of your ~/.ssh directory on the remote side is correct? It should be owned by you, and have 0700 permissions, i.e. chmod 700 ~/.ssh. Also chmod go-w ~ as this is checked also - because anyone with write permission on your home directory can change the permissions of the .ssh directory.

Compare the sshd configuration files (mine's at /etc/ssh/sshd_config) on the old and new server - is something set up differently on the new one, for example does the AuthorizedKeysFile option on the new one point to some other file (I believe some SSH installations nowadays call the file authorized_keys2)?