After spending some time on the XCOM forums and digging through the ini files it's pretty simple to see why the aliens have such an advantage. I played through on Classic the first time through and I thought it was odd how incredibly capable the aliens were.
The developers saw fit to give the aliens a permanent bonus to shooting, and critical hits.
The ability to shoot down UFOs before they get to their abduction and terror sites is actually a feature in the game that is turned off with a simple 0 (and turned on with 1).
You can also control how many aliens are active at once on a map, and a slew of other variables.