Bake brownies by 2 different ways, what changes?
Your first method -- beating eggs and sugar "to the ribbon" -- is a process of denaturing some of the egg proteins into a lattice that will help capture steam as the batter bakes.  Also, this method helps ensure your eggs will be well distributed in the batter, with no stripes of eggwhite anywhere.

Your second method -- beating sugar and softened butter to a pale yellow -- appears to be the creaming method, which uses sugar granules to punch millions of tiny air bubbles into the butter.  By using this method, when the batter bakes, you get a more even "crumb" (texture of the holes in the final cake).  Though you did not mention it, typically this method is used with recipes calling for baking powder/baking soda (you can use the method without a chemical leavener).  Also, I would expect the butter-sugar creaming step to take place before you add your melted chocolate.

Neither method is inherently necessary; if you simply stir all ingredients together at once into a uniform batter, you will still get a tasty product.  That said, each technique will tend to lighten the brownie into a more cake-like texture, rather than a dense, fudge-like texture.  You can do both, if you like, by dividing your sugar in half.