To me a key aspect of playing hickory, especially if you're playing a yee olde period course that's not been modified too much, is that you get a greater appreciation of the course architecture plus how it was necessary to play back in yee olde days.
Matthew says it nicely above when he mentions the intricacies of play, the dearth of spin, modestly lofted niblicks with minimal bounce, landing balls short and feeding them into pins using the ground contour.
As to the original vrs replica/reproduction debate, well there are differences, less so (if at all?) in the woods but modern made irons have better manufacturing and better grooves, some even have a bit of cavity back! But it's the wedges where the main difference is with yee olde originals having limited loft and limited bounce. Once upon a time it was necessary for even better players to treat hazards as hazards and play away or around them. Having a thin bladed low lofted niblick in your hand to play a bunker shot makes you appreciate pretty quickly that modern bunkers (and modern 1st/2nd cut rough) with modern clubs are no longer proper hazards. There was a Feature Interview with Ari Techner a month or so ago where this was discussed -
http://golfclubatlas.com/feature-interview/feature-interview-with-ari-techner/I hope that Sandy's example is taken forward by others, even if only to a limited extent. Imagine, wishful thinking that it may be, that at next years Masters Par-3 contest all the players play with hickories.
atb