Traction on any 2 wheel drive vehicle can be improved with appropriate ballast. ZTRs with little rear overhang and a seat that is mostly in front of the rear wheels have poor traction. I would suspect that your neighbors simplicity either has the seat or the engine or both more rearward compared to your ZTR and this is the reason it has better traction. A simple fix is to adding an appropriate amount of weight as low as possible and behind the rear wheels. What amount of weight is appropriate is something you will have to determine. Adding weight behind the rear wheels counterbalances the weight of the deck, frame, casters and your body weight, which is mostly in front of the rear tires. If your ZTR is balanced poorly to start with, a substantial amount of weight may be needed. However, too much weight will cause the front end to get light. It may not be practical to add as much weight as you need but it's a cheap thing to try if you have some exercise plates lying around. Try adding one 50 pound plate to the bottom of the frame as far back as possible, test and go from there.
If you do trade your machine... I haven't tried the simplicities. But Laser Z's after 2009 have great traction (they were good before 2009 also but even better after 2009)