1969CoronetR/T is "probably" correct on the California emission idea. The bigger hose from that breather is the "inlet" for air that is being sucked through the crankcase (via vacuum at the PCV valve on the other valve cover) so piston "blow-by" gases are burned (someone please correct me if I'm off-base here).
I "suspect" that the smaller line on the same breather you're talking about is there to burn vapors coming off the fuel in your tank too. That's why is sure sounds like a California emissions setup.
So..... No, you don't "need" it as long as your gas tank is vented somewhere else. If you're trying to stay original and/or really like the trees/grass, then leave it on there. Just make sure that if you don't use it, your tank has a vent line on it (or leave the hose going to the breather un-hooked & un-plugged) OR use a vented gas cap. I've heard that without some type of vent & a big fuel demand you can "suck in" a fuel tank, though I've never seen that myself.