Having played both in the last few weeks, it is in fact, much closer than many might think.
Wind, Shinnecock has plenty, but Hutchinson KS is one of the windest places in the USA. I have played PD in 25 plus MPH wind from both the SW and NE directions on my 2 most recent visits, this along with mutiple exposed and elevated tees and greens makes the wind a major factor. During the Women's Open they had little if any wind, which is rare.
Land contours, both greens and fairways, slight advantage to PD in fairways, larger advantage at the greensite. PD's land movement is that of smaller waves, holes 8 and 9. Shinnecock's are massive waves on holes 9,10 and 18. Shinnecock has greens with slighty more pitch, PD's have as a set some of, if not the best internal contours that I have ever seen. "Maxwell Roles"
Best holes at PD: #'s 2,8,9,10 and 17
Best holes at SH: #'s 11,14,16 and 18.
Best par 3 of all 36-#2 at PD, runner up #11 SH
Best par 4 of all 36-#8 at PD runner up #14 SH
Best par 5 of all 36-#17 at PD ties #16 at SH, next best #5 SH
Most indifferent hole at PD #15
Most indifferent hole at SH #8 or maybe #17, these all are still fine holes
PD wins the front nine big, SH kicks butt on the back nine and wins maybe 1 up overall.
Shinnecocks fairway lines are a perfect balance of width (pre USGA Open setup) Prairie Dunes seems to need a bit more width given the penalty for missing the fairway. On a windy day it is much easier to lose a ball or two at PD than at SH, thus the risk of a big number is greater at PD. If you follow the classic archies and their thoughts on the IDEAL Course this is a slight negative PD vs SH. SH's slighty wider fairways also lets one attempt to play the preferred angle of attack on ones second shot into the greens.
All in all, 2 Great courses, much closer in their merits than many might think.