In some patents the employee-inventors are listed together with the institution as assignees. What difference does it make for the inventors to be listed as assignees?
An assignee owns (at least some) rights to the invention.
Does this give the employees-inventors assignee rights at the same level as the institution?
Not necessarily. The actual assignment agreement (ie, contract) governs how the ownership is divided.
If I am preparing a patent as inventor in my company should I claim to have my name listed as assignee together with the company's name?
"Claiming to be listed as assignee" is irrevelant unless you *are* the assignee. Typically, an employment contract governs whether inventors are required to assign all rights over to the employer, or whether inventors retain some rights in the invention.