Using apt installs/updates deb packages from the preconfigured apt repository (which for the Webmin suite is TurnKey's own apt repo). Using the "Install Updates Now" button within Webmin downloads a archive direct from upstream and installs it using Webmin's own internal tools.

So to answer your question, there probably is, but I'm not sure what it might be... I imagine that the guys will be looking to update Webmin and all it's ,modules prior to the v13.1 release (which would mean that the apt-get upgrade should work) but not sure when that will happen.

You need to update your sources.list file to refer to the repository you want to install from.

So you have 3 choices:

1) Stay with Turnkey default and await 13.1 upgrade

2) Modify your sources.list file to point to the Debian backports

3) install the latest webmin directly

NOTE: option 1 is the least risk and option 3 is the most risk (I have upgraded to the latest webmin without issue but this is a caveat emptor comment...i.e. try at your own risk...clone the VM and try on a copy...see how it goes)

See http://www.webmin.com/deb.html for instructions.