Grasshopper, time to learn about Auto Focus, F stops and life
As I mentioned before, your lens is f5.6 at the long end, the 80-200 2.8 is as you think, 2.8. AF systems need light to work. The F stop range works on a doubling (or halving) of the light at each stop. So F4 allows half as much light through as the stop before it, f2.8. F5.6 allows through half as much as F4, or viewed the other way f4 allows double as much light as f5.6. In the comparison of the two lenses here, the f2.8 allows
four times as much light through as the 5.6.
The Auto Focus system works while the mirror is down and diaphragm is fully open, eg at the largest opening (smallest f number). So when your 80-200 is set to f10, the AF works first at f2.8. When your 70-300 is set to f10, the AF is working at F5.6, with
1/4 the light the 80-200 provides. No wonder it works slower, you are making it work in the dark by comparison! You have probably seen this for yourself if you use the Depth Of Field Preview button on the D70 (located under the lens within easy reach of your index finger when your hand is on the handgrip). As an example set your lens to its largest numerical f stop, eg f29, etc. Look through the viewfinder then press the DOF button. You will see the viewfinder darken up, which is the real amount of light getting to your sensor. So now you can see why a faster lens also focuses faster
Life. I hadn't mentioned this and I hate the fact it is so, but like many things which are fun in life, you often have to pay to play. Digital SLRs unfortunately are one of those things. That said many can get excellent results with the 70-300 by applying good technique, shooting on sunny days and working within its limitations. It is just easier and able to handle more difficult conditions with a 80-200 or easier again with a 70-200 which also has AFS (motor in the lens).
Good luck.