Both are virtually the same except for that the Climber has the corkscrew and the Super Tinker has the philips screwdriver in the back layer.
Personally, my choice will always be the Climber aka Super Spartan (get it?) in the 3-layer SAK category (considering only the models currently still in production). I feel the philips screwdriver in the back layer is awkward to use. I prefer the inline philips screwdriver as it feels like a normal screwdriver design (long handle, inline). Also, the can opener tip is small enough to be used in place of the philips screwdriver. Having the corkscrew in the back layer also enables you to carry the mini screwdriver that is very useful in many situations (like fixing eye glasses, small toys, etc).
Anyway, whatever you prefer, you can't go wrong between these two amazing SAK models.
Comparison Photos: Climber (left) and Super Tinker (right)