What happens if air compressor isnt strong enough?

It will work fine until the tank pressure drops below 90 psi. If you keep running the tool below that, it will not perform as well. If you pause for the compressor to refill the tank there will be no problem. With only an 8 gallon tank, there may not be much working time between pauses.

Impact DRILL??

Or do you mean pneumatic impact wrench?

When you have an underpowered air compressor, the pneumatic tool will use more air than the compressor can make, so the pressure will drop below 90 psi if you use the tool too long.

So then you stop and wait for the compressor to build up more air in the tank.

If you aren't using the tool continuously, it won't matter.

I bought an air compressor in 1972 for home use, but I was working on cars.  It was a 240 volt, 2 hp, 2 cylinder, 20 gallon Craftsman.  It was about 7 cfm.  I set it to shut off at 125 psi.  It powered every air tool I ever used on it, including a D/A sander, which really used the air.  It did not like powering my small sandblaster.  It's still at home and it still works, but I rarely use it.

When I moved into my shop, I bought a 240 volt, 3-phase, 5 hp, 2 cylinder, 80 gallon tank.  It powered everything.

-Engine overhaul mechanic and general automotive mechanic since 1972

cfm = cubic feet per minute. That is what the compressor itself can produce continuously. If you allow the compressor to pressurize the tank before you use the impact drill, it should be able to power the drill for limited periods of time by drawing from the tank. I am not motivated to figure out the math regarding how long.

All that will happen is that you will run out of air because the compressor can't keep up.

Make sure you adjust the air regulator to the pressure recommended for the drill.

A 30 gallon compressor with 6 cfm at 90 PSI wold be my minimum choice to power that drill if you're going to use it a lot.

The tank will release the air at a rate much higher than the 3.7cfm so it will run the air tool without issue. The trouble is the 3.7 compressor will not refill the tank as fast as the 4.0 tool empties it. The good news is you run most tools in short blasts not continuously. The tool uses 4.0cfm while it runs non stop but you don't use it non stop so the compressor will have plenty of time to refill the tank. 

Air sanding tools, painting and sandblasting are activities that will run in a non stop fashion. The tools will run the tank empty and then you are relying just the cfm from the compressor. With too little cfm you'll find yourself stopping your work and waiting for the tank to refill. 

The cfm is really the more important spec on a compressor.  Your impact drill will likely be marginal in performance at best.  I have used paint guns where the cfm of compressor was about the same as the paint gun requirement and the gun did not work as advertised.

cfm = cubic feet per minute

the tool might not work at its full capacity.  3.7 and 4.0 is pretty close, you might not see a problem or you might.