What diff are we talking about? The ball diff needs to be built as per the manual with all the right washers in the right positions and you need to use Tamiya AW grease in it (forget the ball diff grease, that doesn't last).
What wheels are you using? Suzuki Swift? They are a brittle plastic. Get some of the older wheels if you can or aftermarket, they have a softer plastic that doesn't break that easily, or Mini wheels (not ideal, again, but better).
You don't need harder gears, the kit ones are perfect. Your diff was slipping big time (if it's a balldiff you didn't build it right, if it's a gear diff, you didn't build it right).
The gear diff shouldn't slip anyway. The only possibility is if you didn't do up the diff screws properly. Use Tamiya AW grease in it and shim if you want, but not that necessary.
There is also the possibility you stripped the outdrive inner splines. Ccheck and address cause (what driveshats do you use? dogbones? Cv joints?- you may need to use the little foam discs to keep the outdrives fully in and engaged with the diff gears.