The peat is stronger in Quarter Cask, both on the nose and in the mouth. QC tastes young and is probably bottled at the perfect age for maximal peat. When peat monsters are bottled too young, they sometimes get unpleasant vegetal notes (i.e. rotting lettuce), but QC is just old enough to avoid these (mostly). CS tastes a few years older, and the peat has mellowed some.
For flavors, QC is straightforward with big peat and big oak, but not much else. CS is more complex with additional vanilla, molasses, and spice notes. The bourbon cask aging clearly imparts more to the CS, even if the overall "woodiness" is slightly stronger in QC. I do like the effects of the quarter cask aging there, but it is not a perfect shortcut to true aging and comes off as slightly bitter (like over-brewed tea).
Both bottles are non chill-filtered and have similar mouth-feel. While not cask strength, QC is a hefty 48 %, which seems to me to be the perfect strength. CS is much too hot out of the bottle, and I end up adding water until it's probably in the neighborhood of 48 %.
QC is younger, maltier, and peatier, while CS is more complex, mature, and balanced. The base Laphroaig peat and malt character is the same in both though, and altogether it's a close match. My advice would be to prefer the Quarter Cask if you have had neither. However, after becoming well-acquainted with both, my money for a next bottle would be with the Cask Strength.
Other Laphroaig reviews:
The Whisky Exchange.