I've owned both at the same time but only kept one of them - the R34GTR. However, that isn't to say that the Z33 is rubbish. Instead it all depends on what you want from the car.
Are you after something you are able to drive daily without hassle in traffic jams, dealing with stop-start driving on your way to work while having enough power to have a little bit of fun from time to time yet still be practical enough to let your wife/mother/sister drive it down to the shops (supposing you bought the automatic or even the manual for that matter) then go for the Z33. You have a huge amount of parts available both looks and performance to select from and you don't necessarily have to go for Japanese brands as the Americans have also explored the limits of the Z33 and with it came improvements and upgrades.
On a day to day basis, I'd say the Z33 is cheaper to run as the fuel consumption would be better than an R34GTR however when it comes to renewing your roadtax each year you'd end up losing all that fuel savings. Having said that, the car is newer and probably cheaper to purchase but the downside is that the market is somewhat more saturated with Z33s for sale. Open any issue of Motor Trader or On The Road and you'd find more of the Z33s for sale than you would BNR34s. Of course, nearly all BNR34s would have to be bought with cash or ridiculous financing.
The Z33 is impractical if you're into golf, have a family or need to carry more than one passenger as there are no rear seats, the glovebox is behind the passenger seat and the boot is so tiny I once bought Logitech Z5500 speakers from Digital Mall and couldn't get the box into the boot - I had to return the next day in another car to collect them.
In stock form both would be equally reliable. You'd be lucky to find totally stock examples of either however the BNR34s would on average come with more mods done to them. Anyway, go test drive at least a couple examples of both the BNR34 and the Z33 only then will you know which 'fits' best.