If you email me more details, I can help you with that problem...I did a couple thousand of those LRA / Louisiana Road Home comps you are referring to....so I can help you get a handle on what your home could sell for now, versus what it was worth before the storm.
Feel free to email me (firstname.lastname@example.org) with the address of the subject property, (your home), and I will send you some reports to better help you with your dispute.
An appraisal is an opinion based on a compilation of information derived from various sources, such as: the Multiple Listing Service, tax records, county courthouse records, etc. The appraiser weighs the worth and reliability of each piece of information, and also uses various methods of approach, to come to his conclusion.
Yes, facts are involved, but so is the human approach; therefore, appraisals from various sources will vary.
Also, I don't really understand who the "they" you cite in your question is, but if you are comparing an agent's appraisal to a professional real estate appraiser's appraisal, know that they will be different. An agent's appraisal is really an informal "comparative market analysis" that uses data from recently sold comparables. An appraiser uses various approaches + facts in order to come to his value.
To find out what homes in your area have sold for, you can look up the tax records online or contact a local Realtor to compile a list of recently sold comparables for you.