I have a vps running ubuntu and set up a subdomain that forwards to the vps' ip address. I thought this would be an easy way to connect without having to memorise the ip address.

Now when I connect to the vps with PuTTy SSH using the ip address all works fine. When I connect using the subdomain I get the log in prompt but my log in doesn't work. I noticed that the "source" after the '@', following the 'root' differs:

login as: root
root@12.34.56.78's password:
vs
root@mysubdomain.mydomain.com's password:

Is this the problem? Can I fix this somehow? How do I do this properly?

In general, it doesn't matter whether you use symbolic name or numeric address provided symbolic names resolves properly (does it?) but it may be a factor in certain specific sshd configurations e.g. if UseDNS is enabled and reverse lookup fails on the server or if Match or AllowUsers rules are set or if there is a mismatching record in known_hosts, etc, etc. In short, there are many reasons it can go wrong and the best way to find out which one is to launch sshd on the server with verbose or debug LogLevel and check the logs. man sshd_config.

PS Opening remote access for root is not a good idea. Also for ssh forwarding is something else and is not related.