Generally speaking, Bishopstown is probably considered more upmarket - used to be called "snob alley", but some parts of Bishopstown are a bit "rough". Turner's cross is probably more mature and much of the area is council or former council housing. A very high proportion of former council housing is now private.
Roads: Bishopstown into Cork can, very often, be an awful drag, with very heavy..and slow traffic. Turner's cross to town is much quicker.
Walking: It will take you 10 mins to walk into the City centre from "De Cross", whereas B'town will take you up to an hour.
Subsidence: BIG problem in B'town - you may not be able to get insurance cover for it.
Schools: "Good" schools - lots of crap about that in Cork (and elsewhere) There are several "good" schools within walking distance or a short commute of Turner's X - you have a wider choice than B'Town, though equally any you might want to use are within easy commuting distance.
Parking: Tighter in Turner's X and, when there are matches on the place can get very heavy. Bishopstown is very heavy when the CIT, in particular, and UCC are in session.
Parks: The Old city dump, now closed, is due to open shortly as a public park, so this will be right on your doorstep in Turner's X - this covers something over a square mile, probably more.
Access to South Link/N22: Not bad from either B'town or Turners' X but probably easier from the X.
Shopping Centres: Reasonably near to several in either location
Up de rebels!!