These two words have different meanings: 'Reimburse' usually involves simply getting back money which you have spent eg you paid to travel somewhere for your work, and your company reimburses your travel expenses. 'Compensate' , by contrast, always involves the other party admitting liability of some kind. They are at fault, and - by way of apologising to you for the damage done/ trouble caused/ unnecessary expenses incurred - they pay compensation to you. For example if you have an accident at work and it is your company's fault, you will receive compensation for this. I hope that's clear.

You could substitute "compensate" for "reimburse" in the second sentence, but it might change the exact meaning, although the main point would still be the same. However, in the first sentence you cannot. We will reimburse any expenses incurred. Means that they will give the person their money spent on expenses back. Although synoyms, "compensate" has a broader meaning than "reimburse". In everyday English, you will see "reimburse" being used as "refund": If the item doesn't match the description on the website, your money will be reimbursed. Notice you can't use "compensate" in this sentence.