Key Difference:
A Perfectly Competitive firm is a "Price Taker". It can sell as much as it wants at the market price, meaning $P = Demand = MR = AR$ (Mr. DARP).

A Monopoly is a "Price Maker". To sell more units, it MUST lower the price on ALL previous units, which causes $MR$ to drop twice as fast as Demand.

Because Monopolies always have higher Marginal Costs. Because it produces where Price > Marginal Cost, meaning society values the additional units more than they cost to make. Because $MR$ is greater than Demand.