After restoring the db on the other machine, you will probably also need to create a new db user and make it dbo on the restored db and use it instead of the old user in the connection string.
The part about servername really depends on configuration. The \sqlexpress part after the \ is the "instance name" of the MS SQL instance. You can have more than one instance of MS SQL installed on a machine (even different versions) and they would need different instance names. There is also the concept of a "Default Instance" and this instance has no name and is just referenced by the server name or ip address.
The difference in practice boils down to differences in installation defaults. SQL Express defaults to have an instance name of SQLExpress when it installs whereas the full version defaults to be a "Default Instance" unless it detects another default instance already on the machine. But in both cases it is possible to change these during install, you can make a SQLExpress installation without an instance name just as you can create a full MS SQL install and use an instance name.
So the bottom line is when you move it to the other machine with full MS SQL it probably is a "Default Instance" and thus you won't need an instance name in the connection string, but it is possible that you would need to if it was configured differently during installation.
Hope it helps,