Just a couple of thoughts, I'm killing about 20 minutes waiting for laundry.
1) Being properly weighted is not the same thing as being neutrally buoyant. You can be neutral even if you are grossly overweighted as long as your BCD has enough lift to counter the weight. You can also be neutral if you are underweighted, and while I don't find this exhausting, I do find it very uncomfortable, which I will explain in a minute.
2) Proper weighting allows the diver to:
Minimize drag by not pushing an unnecessarily large bubble through the water. In the event that the diver is propelling them self by swimming, this reduction also reduces effort, exertion, and therefore air consumption.
Be more stable in the water because a giant bubble isn't migrating all over their back, one minute at their neck, the next on their butt.
Most importantly in my opinion, it allows the diver to control their buoyancy with their lungs. The greater the volume of a bubble of air in your bcd, the greater the effect of depth changes on your buoyancy. If the volume of the bubble is great enough, even a slight change in depth will cause a change in buoyancy beyond the capacity of your lungs to control. At this point you either have to deal with it by adding or removing air from your BCD. This makes diving complicated, difficult and downright unnatural. Sometimes, like when you dive double 121's with a meaty fill, you have about a 22 lb swing to deal with between the full and empty-ish tanks, and there's just no way to avoid it, and a 22 lb bubble is a pretty big bubble.
Even if you're properly weighted, significant changes in depth will require some operation of your bcd, but the goal of proper weighting is to minimize this.
If you take too much weight off, and you are trying to dive using your lungs to control your buoyancy, you will find that you have to keep them pretty empty, which is an uncomfortable, very dissatisfying way of breathing. I personally breath alot more when I find myself in this situation, and feel that, while being overweighted may complicate diving, being underweighted uses alot more air.
I'm off to wash some clothes.