The choice is about: freedom of movement vs hassle & costs.
The good thing about your own vehicle is that it will take you wherever you want to go, whenever you want to go. Even to the places that are not served by public transport. The downside is that crossing borders with a car increases hassle, this will be most notable when shipping your car around the Darien Gap.
The good thing about using public transport is that is easy and (likely) more expensive*.
*) Although public transport is considerably more expensive per mile traveled, I calculated that the costs for shipping (+handlings and border duties) around the Darien Gap is a big factor in the overall costs of doing this trip with your own vehicle. So driving your own vehicle may turn out to be more expensive. The more miles you travel, the cheaper driving your own vehicle will be. I did my calculations for 2 people, so I estimate it would turn out to be cheaper to drive your own car, if you do the calculations for 3 people.
Most travelers I talked to told me that by selling a car in the south that you bought in the north, you will likely not loose too much money, maybe even make a small profit. (selling a car in the north that you bought in the south is going to cost ya).