Yes, the difference can be claimed, but there's no guarantee that your refund will increase. Here's how this works: you claim the miles he drove using the standard mileage rate, and include any other unreimbursed (or partially reimbursed) expenses that he had for work. Towards the end, you will be asked the amount of reimbursement he received. This will be deducted from the full amount and the remaining figure is the total of unreimbursed expenses.
However, whether or not it will impact your return will be based on two other factors: The amount of the expenses that are greater than 2% of your AGI (as a couple), and then if you are itemizing your deductions. These FAQs answer more on this. First, what kind of expenses are deductible: https://ttlc.intuit.com/replies/3300622
Next, this one explaining the 2% rule: https://ttlc.intuit.com/replies/4800344
And finally, this one shows where and how to enter in TurboTax: https://ttlc.intuit.com/replies/4800418
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