A pergola has an open or louvered roof. A patio cover has a solid roof. They serve different functions and price out very differently.
Pergolas are about controlling sun while staying open to the sky. Modern aluminum pergolas use motorized louvers that pivot from open to closed in under a minute. Closed, they shed light rain. Open, they let the breeze through. They do not make a room outside; they make the outside more usable.
Patio covers, by contrast, are extending the conditioned envelope of the house. Solid roof, sometimes insulated, often with integrated lighting and ceiling fans. The cost is meaningfully higher because permitting is closer to a room addition than a landscape feature.
For hospitality projects, we usually recommend pergolas for restaurant terraces and patio covers for poolside cabanas. The decision usually comes down to one question: do you need to keep furniture and equipment dry year-round?