Today's marching snares are quite a bit different, mostly to withstand high tension tuning and Kevlar heads. Older marching snares were similar in construction to drum set snares, but tended to be set apart by the following differences:
Marching drums are generally deeper than drum set snares. Of course there are a few exceptions, but you normally won't see a drum set snare deeper than 8" or a marching snare shallower than 10". (Unless maybe you're in Europe, where shallower drums have been used for marching in some places.)
Marching drums often have catgut or synthetic gut snares instead of the wire snares usually found on their drum set counterparts. Really old drums could have wire-wound silk snares that look a little bit like wound metal guitar strings.
Starting in the 1960s or '70s, marching snare tuning started creeping higher and higher, so you'll see marching snares with heavier-duty shells and hardware, particularly on higher-end models. The closer you get to the present, the beefier that hardware tends to be.