A professional, independent appraiser ties up all the loose ends of an important purchase. The appraiser makes certain that the diamond is in the condition listed on the report at the time of delivery to you rather than months or even years later. If there is anything wrong, the only time to know it is right away. The appraiser confirms the diamond is the correct diamond matching the report, too. It is not uncommon to see 1.00 carat diamonds getting mixed up in large dealer inventories. If your diamond is unique, then that is less likely to happen.
Finally, the appraiser arrives at an independent opinion of value, confirms that the stone is skillfully set, in a properly constructed mounting and that any remaining questions and concerns you may have are all answered. If you do have a loss, the independent valuation of an independent the appraiser is an excellent tool to force a fair and proper settlement or replacement from your insurer. Poorly constructed appraisals may allow your insurer to corner you into accepting a poorly matched replacement.
You might want to get an appraisal simply because you distrust the seller, but there are better and more substantial reasons that consumers need a second and unbiased opinion. It is easier to cure a problem when the deal is very new than months or years later when it has become your fault and way too late.