It's all about reducing drag and as TT bikes prove, you can make a flat bar bike very aerodynamic. However, most flat bar bikes are hybrids and the geometry is all wrong to get good aerodynamics. A true flat bar road bike can be made as aerodynamic as a drop bar road bike if the handlebars are low enough, narrow enough, and far enough forward. How far you want to push it versus what's comfortable is up to you. Also, what % difference it makes will depend on your air speed. At low speeds, it doesn't make much difference. At high speeds, it makes a lot of difference. Way too many variables to quote accurate numbers. In your case, it could be 2% or it could be 20%. It depends.

On my way home tonight, I passed someone on a beautiful CF flat bar road bike. Turned out to be a Pinarello Prince with what appeared to be 58cm bars. It even had a bar end mirror on it. I have nothing against flat bar bikes, but that was like painting a mustache on the Mona Lisa. So sad...