tldr: it's not that easy.
I'll take a stab and Bluto can correct or jump in where he wants to. Without knowing the specifics of the purchase, it's hard to 'explain' the discrepancy between the appraised value and purchase price. With commercial properties, there are all sorts of intangibles that are not taxable that are typically included in a purchase. It's why Walgreens pays at least 2x for a property on a corner at a busy intersection. The property isn't worth that much, but it's a Walgreens business decision. Then you have competitors trying to buy up land parcels to keep competitors out and overpaying for the same (see Lowe's and Home Depot acquisitions or even CVS and Walgreens). Those overages are not taxable because the properties are simply not worth that much on the open market. Then you can get into other avenues like 1031 exchanges where companies or individuals simply need to park their money to avoid a tax consequence who simply do not care about overpaying. The list goes on and on. In other words, it's not that simple and we'd need to know specific facts about the purchase to try and understand why TCAD landed on that valuation.
All of that being said, you're focused on the market value and not the equity value. If all other comparable properties indicate a value of $78mm, then it will be on the tax roll for $78mm. To get to 108mm, the District would need to raise the entire asset class by 38% to reflect a $108mm value. But does 1 sale make a market?