Is it permissible for an employee to keep the financial difference between the housing allowance allocated for travel ($500 per day) and the actual lower cost of the hotel, given that the company requests hotel invoices?

If the company grants its employee a sum of money for housing by way of ownership, then he may take the surplus for himself. However, if the amount is by way of agency for him to spend on housing, then he must spend it in full, and he is not allowed to take the surplus except with the company's permission. The company's request for hotel invoices makes the second possibility more likely. If the employee submits the real invoices and the company leaves him the surplus, there is no blame on him. Otherwise, he must return it, and he is not allowed to resort to trickery with fake invoices.