Direct copy from Daniel Stearn's website >>
WHY USE RELAYS?

Power for the headlights is controlled by a switch on the dash. This is *not* a great place to tap into the system, for two reasons: The headlamp switch uses tiny, high-resistance contacts to complete circuits, and the wire lengths required to run from the battery to the dashboard and all the way out to the headlamps creates excessive resistive voltage drop, especially with the thin wires used in most factory installations.

In many cases, the thin factory wires are inadequate even for the stock headlamp equipment. Headlamp bulb light output is severely compromised with decreased voltage. The drop in light output is not linear, it is exponential with the power 3.4. For example, let's consider a 9006 low beam bulb rated 1000 lumens at 12.8 Volts and plug in different voltages:

10.5V : 510 lumens
11.0V : 597 lumens
11.5V : 695 lumens
12.0V : 803 lumens
12.5V : 923 lumens
12.8V : 1000 lumens ;Rated output voltage
13.0V : 1054 lumens
13.5V : 1198 lumens.
14.0V : 1356 lumens ;Rated life voltage
14.5V : 1528 lumens

edit: The 12.8v at the bulb(a common scenario with old stock wiring) upgraded with relays and heavier wiring could likely reach the 14.0v level at the bulb if driven directly from the alternator. This is a a 35.6% increase in RATED LIGHT OUTPUT which is usable light. I think you'll feel the difference with 1/3 MORE LIGHT. Relays alone will do wonders. Halogen on top is icing on the cake.

An example like using a specific halogen light so you can see how much more you get would be interesting