First off, the elise will be a better balanced car. The TT will be nose heavy, and engineered to handle 'safely', i.e. understeer, (as most modern cars). For the skilled driver IMHO oversteer is prefferable
The elise will also handle better in the corners, especially while being subject to gentle of full acceleration, as the weight will transfer to the rear wheels, giving them excellent traction to propel the car, leaving the front wheels to control the the direction. In a 4x4, the front tyres are having to cope with acceleration & steering requirements, therefore, asking more of the tyres will cause them to loose traction sooner... leading most likely to understeer

The key to it all, is to get the liz nose heavy under braking, then, just after you come off the brakes, while the weight is still over the front, get the front end turned into the corner, as the weight transfers to the rear, you can get on the power, and 'drift' the car, beautifully balanced around the corner. Too much power and the rear will start coming round, so you can adjust that with the throttle, & / or steering. 4X4 will be much less able in that circumstance. It will tend to understeer, where more power will push it wider, off line.
The elise's greatest trump card over the TT is it's lack of wieght. A TT at 1260 - 1520kg (dependant on model) has much more weight to haul around a corner than the Liz(860kg). The laws of physics take over! Momentum will carry the already understeery TT off line, while the light nimble Liz, with it's lightweight, super stiff chasis, will grip, and go round the bend.
The only advantage the TT would have, is when you are sliding, it has all four wheels to pull you along... if you can get all four pointing where you want to go! It will have more traction in the wet at the traffic lights grand prix.
Ultimatly, the Liz would be quicker across any country road, and would by my choice. IMHO
TTs of old were not highly regarded as driver's cars. Too heavy & slighly numb.