I think this is a superb question and I know Tom is looking for decisive/persuasive information.
All I can offer is this:
Sleepy has got the...memorability factor. The views, the variety of sometimes "audacious" holes, the CBM-Raynor-Template-Engineered cut of its jib, the scale of the property (which, however, can be a negative to the walker - it's a rough walk). I think the Sleepy round lingers in the mind with greater detail and recall than that of Quaker.
For me, Quaker has the three best holes of the 36...#s 9, 14 and #17. Quaker has the three most underrated-sublime holes of the 36...#s 3, 11 & 13 (though 11 gets ink and is oft-cited as a "signature" hole...it's better than even such advertisement.
While perhaps not with the acuteness of a Merion, Quaker is a "neighborhood" course, while not "hidden" from view per se, its got the kind of Wrigley Field charm in that you are in a common (but upscale) neighborhood and boom...there's the familiar logo sign, the English Tudor clubhouse and you say, "Geez, so much history and veneration right off the side of the street." This feeling is amplified immeasurably if you approach the club from the west side of Griffen Avenue when you course along one of Quaker's most audacious holes, the 8th (S. Abramson's 3rd photo below) along your left/driver's side.
Sleepy Hollow feels more remote...out in the country, and there's some nice mystery/zest in that too. And the views of the magnificent cream stone clubhouse on the approach are very inspiring...teeing off and ending in its lording shadow are part of that memorability I've referenced. AND the routing takes you out in the country too...you wind and scrawl over hill and dale across ravines and down slopes and by the time you're on 8 or 9, you really have no idea where you are in relation to where you started and how you are going to wind up on the 18th tee. For me this feeling lingers even upon repeat playings, albeit less than that first time where I truly was "lost." This perhaps sounds more appealing, but it really matters as to temperament about such things.
I think the most telling analogy is to give these cases of the "12 HCP."
If you are a 12 HCP who's struggling and not playing your game... a top here, a chunk there, a block slice, a second chili dip, a disinterested 3 putt or four...both courses are going to eat you up, but you're likely to accept the experience with more amusement at Sleepy than at Quaker. At Quaker, that guy never breaks 90 and is more sour
If you are a 12 HCP who has a superb day and breaks 80 from a 6500+ tee, you're going to really know/think you did something at Quaker and you're going to mention it every time you encounter a friend-golfer who doesn't know what you shot there. On such a day, you're going to remember the "round" and therefore, the course, with greater respect than if you have such a day at Sleepy, which though every bit as difficult "feels" (as many template courses do) that it was available if you could hit the ball mostly where you wanted. Quaker is more sublime and makes you--as a golfer--feel more sublimely "worthy" of that good score....which also leads to the 3rd case of the 12HCP
If you are a 12 HCP who plays his normal game and shoots 82-86, you're going to want to return to Sleepy moreso than Quaker because you know better was available. You "knew" that you should've taken more club, you "knew" you should have trusted what the caddie said about aiming on a blind shot, you "knew" you should've not feared that long putt, you "knew" it was more open on this side than the one.
This last case is another point of "memorability" that is not always understood (and another point of the appeal of template courses)...Though we can't always control our golf ball on command, and blindness is incorporated into some of the template designs...they seem to have a unanimity in their obvious commands...you know what to do at a Cape...a Redan...a Short...a Punchbowl...you know what the decision is on a Leven...a Sahara...double plateau...you know where not to miss on an Eden, a Road, a Prize-Dog-leg. Depending on your game and temperament, there's a delightful frustration in that - when it was obvious what to do, but you can't do it with any consistency.
Meanwhile at Quaker (and a lot of the Tillies I know) you've got to seek out the correct strategy shot by shot...While some shots are plain an obvious, there's a bit more figuring out your own path and compensation for imperfection, you may understand some Tillie-generalities..."don't miss to the sides of greens"..."club to the front edge" "stay from 8 to 4 o clock. Tillie's are somehow more frank, yet more camouflaged shot by shot.
That advertisement for my subtle brilliance aside, I have to give it to Sleepy by more than a nose, but less than a head...unless I have to walk, in which case I'll be at Quaker.
cheers
vk