The biggest whopper of a reason? Your images will be sharp and crisp. In photography, yes, it's all about the light, but it's just as much about intentional focus. That means things that are supposed to be in focus are clear and sharp as a tack. By using a tripod, you eliminate a lot of the shake that comes with shooting freestyle. Shake = blur.
Your interior shots will have straighter lines. When you use a level (most tripods have this built into their base), you can square up your vertical and horizon lines while you're photographing a room. This can elevate your images about 10 fold, this precision. Next time you're flipping through Dwell or Elle Decor, notice the where each room wall meets to form a corner, you can bet a dollar that line will be perfectly straight and parallel to any other vertical lines. Same with the horizontals. Go and look! You know who defies this convention often? Todd Selby. He's a rebel and he rocks, that's a whole other story.
Another thing, you can shoot from a lower perspective. When shooting at eye level, you are off axis to a lot of the furniture in any given room. Plus, as viewers we pick up on the photographer shooting at eye level and we recognize it. When you get lower, more perpendicular with major pieces in a room, you achieve those straighter line I'm going on about and you shoot the furniture more head on. By using a tripod and setting your camera at about stomach-chest high, you get that perspective.
I've included a few images from a tour I shot to illustrate what I'm talking about — the photos are crisp, the lines are straight up and down where one wall meets another and they were all taken at my chest level (I'm pretty short, 5'4", so this height may vary for taller and smaller people).