Re: How important is it to use "trailer tires"
Some tire makers use the same tire for truck and trailer (literally the same tire), so you know there is zero difference in those two. Passenger tires (note they are not actually called 'car' tires.. for a reason) are more shock absorbant tires, which makes the ride of a car softer but is not normally a plus on a trailer (causes more rolling). Passenger tires are perfectly suitable so long as you buy tires rated at 115% of the weight you are towing (the extra 15% is to handle the added stresses of corners and bumps).
As with all things, some variations are better suited for certain uses. If you won't be approaching a passenger tire's load rating, there is no requirement to buy a trailer tire rated even higher. One is enough, the other may be overkill. After all, using a trailer tire rated for 4000lbs on a 500lb towable will likely beat the towable to death, as there will be nearly no give for bumps. The opposite being using a passenger tire rated at 4000lbs for a 4000lb load. The towable will tend to bounce and roll. There is a sweet spot (~85% of rated load) and the closer you are the better, no matter the type of tire.