Just to check: you are using the public Main Galaxy server at http://usegalaxy.org?
This may seem like a simple question, but did you log into your account when using Galaxy from the second computer? The same account?
Doing this will close all other active sessions associated with your account and the current session will be active. This should be identical in content from when you logged in previously - from any location.
If you want to move data between your own Histories, try the functions in the History menu such as "Copy History/Datasets".
As Martin asks, please provide more information if this is not your situation and proper login was not the problem. Best, Jen, Galaxy team