Pureeing and Straining
To puree is to use a blender, food mill, etc to make a sauce smooth. To strain is to take a liquid (pureed or not) and pour it through a fine mesh to remove small particulates and make it completely smooth.
Pureeing and straining are two largely optional steps in sauce making, but a pureed sauce will be more refined than one that is not pureed. And a pureed and strained sauce will be more refined than one that has only been pureed. Refining just refers to the number of steps it takes to get to a finished product.
If you are serving a rustic meal, a rustic (unpureed, unstrained) sauce might be appropriate. If serving a very special meal, you may want to pair it with a more refined sauce.
To make a completely smooth and velvety sauce, a chef knows that, if a pureed sauce is smooth, a pureed and strained sauce will be either smoother. Now you know, too.