The North Downs.
Seriously, that’s two questions really, because the N.Downs and the Surrey Hills are two discrete but very close areas.
The N.Downs is the ridge running E-W and in the area you’re talking about I guess you mean from Guildford over to sort of Ranmore / Boxhill. It’s more clay based there and will be pretty shitty (plus the 4×4’ers use large sections of the NDW which you can’t avoid totally if you want to cover any distance)
The Surrey Hills run more broadly N-S, and I think of mainly as 3 ‘doorwedge’ shaped hills, climbing steadily from the A25 to reach their heights at Pitch / Holmbury / Leith, and then dropping sharply off the tops of these onto the flatlands beyond. Hence the climbs from the A25 to the top of any of these are generally longer and steady, but go W-E at the southerly end and you get shorter sharper climbs. This area is more sandy and drains far better, will still be wet but less claggy.
If it were me I’d stick to the Surrey Hills.