what would happen if you hooked a car audio sub up to a home audio amp?

The DC-AC differences really only effect the way the amp produces power.

What is delivered to the sub/speakers is essentially the same.

The main difference is home audio amps are usually meant to power higher impedance speakers than car audio amp.

A home amp will most often be specified for 8 ohm, or in some cases 6 ohm loads. I've only seen subwoofer specific amps capable of 4 ohm loads.

It's just a matter of making sure the impedance match up is acceptable.

Yes but the impedance is wrong from car systems to home and it wont sound as good or it could blow it if cranked up to high..