I agree with dbrock34. Just having the heavier rig helps with the towing aspect, plus bigger brakes, springs, shocks, typically heavier duty cooling for towing grades, etc.
If you put 6k behind a 6k truck, it's alright. It'll be safe and stable. But if you put 6k behind a 7400lb truck or a 7800lb truck (typically diesels are the heavier), then your truck acts like an anchor and the load doesnt pull your rear end around as easy (bow wake from 18-wheelers for example), or push and porpoise your back end with the beefier springs and shocks.
But on the other hand, like Vern stated, a 1500 is plenty of truck for that same 6k. I wouldn't hesitate towing with the correct setup on it.
My two cents: I went with a 2500 power wagon because even though I only tow around 8k max, I wanted my truck and cargo to weigh more than the load I was towing. Just felt more secure, rock solid, stable.