Technically, 80 lbs cover and matte (eg. 10pts) are usually the same unless it's specified as coated (eg. glossy on one or two sides.) That's the stock often used on very economical business cards.
The paper used for standard books, letterheads or on a laser printer is closer to 24-35 lbs matte, to give you an idea.
When they refer to a text paper, even if it's a 80lbs, it's thinner than a cover 80 lbs. It's what you often see used for pizza flyers and most brochures. But if your printer says it's matte, then it has a soft satin compressed feel but it's not as thick as a cover. It's usually very opaque, bright and white.
When you see heavier paper, it doesn't mean it's thicker. The text type of paper is as heavy as the cover but more "compressed",
therefore it doesn't feel thicker.
My preference for a flyer is to use 80lbs text and less; the flaps close well and it does feel like a standard commercial printing without looking cheap. You can always use the 100 lbs text version if you like very thick brochures. I don't recommend any cover type unless you use 65 lbs and less.
If you want something more "natural" then you'll go in the real dull category where you can see and feel more the texture of the paper. Make sure to specify if you want it extra white or not; there's different types of dull, satin, semi-coated and coated stocks used for offset printing and the best is to simply ask your printer and choose a sample.
All stocks have their own brands as well with different brightness, and the weight is just one part of the choice!
Here's a scale with a good reference for which paper to use: