that I can see is the desirability of a surface support team, safety divers, etc. to set up and run a safe dive site.
As Bob3 noted, thirds and redundant air are necessary, but these are prudent when planning deep wreck dives also. They are not techniques or equipment unique to cave diving (or ice). (Perhaps I misunderstand your question.)
In terms of using cave techniques beneath ice, I cannot think of many similarities. They are two very different evolutions. As noted previously, the 'buddy teams' are one wet and one dry communicating via the line. Divers do not dive in close proximity for fear of fouling one another's lines (thus sending false signals). Not like cave penetration teams that I've heard of. (I am not a certified cave diver.)
ABOVE the ice, however, its another matter. We used one team to cut the triangle with chainsaws, a second to shovel the lanes that radiate out away from the hole in all directions, a third to set out wooden pallets & tarps, insert ice screws and lay-out safety lines, and a forth to set up the warming tent and start the stoves. Having organized support personnel makes all the difference in the world. While I've never been involved in any organized cave diving logistics (e.g. WKPP, Mexico, etc.), from all I've ever read or heard of them this is quite similar - the surface support staff and safety divers make it go more smoothly and safely for the penetration divers.
Thats the most striking similarity from my perspective.
My $.02 worth!
Doc