... he said that he had trouble finding comp sales in the neighborhood and he would probably have to find some from the next town over from us, about 7 miles away.
The most common misconception is that "value" is a characteristic that can be objectively measured, like weight, height, size, speed, etc. It's not. The definition appraisers use involves an "opinion" as to what the property would "most likely" sell for, with "typical" buyers and sellers.
So appraisals are basically an educated guess, sort of a prediction of what would have happened if the property had been placed on the market and successfully sold. When you look at it in those terms, and acknowledge that there is actually no such thing as a "typical" person, or "normal" reaction to a property, it's easy to see why different appraisers might come to different conclusions. Much in the same way that every sale of identical homes wouldn't result in exactly the same sales price.
Then, in your case we're dealing with even more variance because of the lack of sales data. Not only are the buyers and sellers going to have more variance in their actions, and not only will there be more variance in property characteristics, but different appraisers will have different opinions as to which data is most relevant. For example, an appraiser would have to decide whether a very old sale in your immediate area is a better indication of value than a recent sale in a distant area. And different appraisers will have different opinions regarding how they handle conflicting data.
To address the question in the title of your post, no, theoretically the appraised value should be the same for every appraisal. In the real world, it could be argued that a more thorough appraisal is likely to lead to a more credible value but it doesn't always work out that way. Also, in the real world, some appraisals are done with a wink and a nod, knowing that the appraised value only needs to be some lower, more easily supportable value for the lender to qualify for the loan. Or, the assessor might be applying some pressure for his appraisers to develop the highest possible supportable value to increase tax revenues. It's not supposed to happen that way, but it does. (Given that there value can be rather ambiguous in some situations, many appraisers would prefer providing their opinion of value as a range instead of an exact dollar amount.)
That's why, even with the best appraisal, I recommend looking at your situation in terms of probabilities, risk, worst/best case scenarios, and your ability to handle the risks and maintain personal cash-flow even if the worst case scenario plays out. When it comes time to sell the house, even the most logical, diligent, and well-informed appraiser is likely to have an opinion that doesn't exactly line up with your final deal.