The longer you can leave them on the plant, the more likely they will be fully ripe when you collect them. Depending on the plant, the seed pod is likely to turn brown and look dead or start to crack open when the seeds are ripe.
The main reason not to leave them on the plant right into winter is that the pods are likely to open enough to let the seeds fall out. (And they might get rained on, and rot. Or birds might eat them so you can't save them.)
The type of plant REALLY matters since some plants hold their seeds for weeks, and others ripen and then "pop" overnight.
If they are like poppies, you might need to "bag" the pods before they turn brown, to be sure they don't ripen sneakily, then open and drop overnight. Often people use "organza bags" from a craft store bridal isle to catch the escaping seeds and trap them in a drawstring fine-mesh bag that lets air and rain pass through.
It really depends on what plant you are saving seed from.
