In reply to oldie:
The ban was introduced waaayyyy before my time as Swanage Access Rep. However, I believe that the ban on Tilly Whim was part of an overall agreement to resolve a potential climbing ban in the whole of Durlston Country Park (i.e. Lighthouse Cliffs, Subluminal and most of the Ruckle). I understand that, at the time, it was seen as a success for climbers that we only lost Tilly Whim.
I've not climbed at Tilly Whim, but abseil access would involve climbing over the stone wall by the tourist path directly onto a steep grassy slope, right in front of lots of grockles - probably not a good idea. The old tunnel access could be re-opened, but that would also allow climbers and grockles to enter the caves and possibly disturb wildlife (bats/birds) and be at risk of rockfall (though no worse than Winspit/Hedbury surely).
Note - it's not just climbing that's banned at Tilly Whim. All public access is discouraged and there are signs/walls/fences there to discourage people from entering.
Note also that the Tilly Whim area isn't covered by CRoW.