I tend to pick mine earlier due to the danger of some bug or critter eating into them. But I dont have a definitive answer if whether leaving them on the vine longer improves the taste.

Here is my rule of thumb on this. Look at the vine, is it alive and still growing or dead/dying an brown? If alive, leave 'em a little longer . If dead pick 'em now, get them out of the sun.

Once the pumpkin is picked, or the vine dies they clock starts ticking and the pumpkins drying out. A live vine will maintain moisture levels in pumpkin. But a dead vine will not and out in the patch in the sun the pumpkin will dry much more quickly.

Agree with JC. Wait until all leaves are dead and gone and the vines are shot. Even if the powdery mildew is going wild and the vines and leaves are in severe decline. Let the whole thing wind down in its own time. The fruit stems should be brown, dry and hard assuming it's a Pepo and not a Maxima. If it's a Maxima the stems tend to be corky but they should still be dry and brown for complete ripeness.