It's not widely recognized, but rear end gears generate a lot of heat during use. If you touched the center section of a race car's rear end that just came off a roadcourse you could actually burn your fingers. Many racecars even have coolers for the rear end oil.
When things get hot, they expand. The rear isn't even halfway full of oil, the rest is air. As the oil and air both expand the vent lets air out. If not for the vent, you'be be blowing seals out on highway trips. Later on the rear end cools off, everything shrinks back again and fresh air is drawn back in. So you really need that vent.
The hose is basically to keep water out of the rear. And tucked up out of the road draft supposedly helps minimize the amount of dust in the tube. Since the tube never signifigantly sucks air in, no filter is needed. Cooling down takes a long time, the drawing in of air is a VERY gradual process.
Can you live without the hose? Sure. You'd want to get one on there sometime though.
I'll add that unless you keep keep the opening clear, it'll clog with dirt, mud or whatever and seal the housing off which will let it all build pressure. As Tracy said, pressure will blow a seal because of the the heating of the oil and air. It doesn't take much. About the other end of the hose, there should be a hole in the frame above the hose nipple that the hose slips into. It might or moght not have a grommet, mine does. Make the hose longer to allow for suspension movement and affix to both ends. Also, ensure that the vent hole is open before you install the new hose -- I have used very small drill bits by hand in the past. You can also remove it and clean it but you'd need to reseal it as it hold the brake distribution block on the rear.