For us, Mexico provided the most grief, but we equally had some of best times in Mexico, especially in the east. Mexico is quite a difficult country to over land independently, given the conditions of the roads, the horrendous driving from other drivers, the potential for bribes and the checkpoints. I can understand the appeal of organised tours here, which is rare for me to say. However, I do think the west and north are better seen by having your own vehicle.
Belize, instantly felt a world away from Mexico as soon as we entered the country. We definitely think we won out going to a lot more places than those without a car. Some places were also only accessible by a pricey tour or your own vehicle. Due to this, we often found we had places all to ourselves. Unfortunately, we heard on the grapevine that renting a car here isn't cheap (like the rest of the country), so you'd have to way up the cost of hiring versus a tour to see places like Cockscomb Basin and exploring the Mountain Pine Ridge Reserve.
Guatemala is definitely a country, where going on chicken buses and collectivos wins out. They're cheap, regular, and cover plenty of places. Plus you don't run the risk of ruining your vehicle like we did on the Rio Dulce to Semuc Chempney road. Other than that road, we found the highways to be good though.
Honduras seems to be best tackled with your own vehicle. We heard numerous reports from other travellers about the safety of buses (or rather lack thereof) and having to go through the ever unpopular city and most dangerous in the world, San Pedro Sula. We had a lot more flexibility than those without their own vehicle and again, we were able to go places the buses didn't frequently venture, if at all.
Nicaragua has some of best roads and is easily accomplished by having a non-4WD vehicle. Again, we managed to fit in a lot more with having our own vehicle, but if you don't choose this option, there are plenty of buses leaving from major towns (you just won't fit as much into your itinerary as you would with having your vehicle).
Costa Rica has surprisingly a raw deal on the road front and there are parts that a 4WD is needed eg the area around Corcovado and further down the peninsular. Like Belize, this isn't a cheap country to travel, so way up the need for a hire car, if you don't have your own vehicle already. Buses took double the duration it took us to get anywhere and you are stuck to their schedule (often very early in the morning). If you are just here for a holiday and don't fancy the stress of some of the roads if you are going to the above places, then you might want to consider a tour out of ease.
Panama is easily tackled by buses, which are cheap and tend to have a regular schedule. Every place we went to has good transportation links, so your own vehicle isn't necessary. It's also more of a hindrance driving a car around Panama City, which is nothing short of chaos during the many rush hours throughout the day.
Overall, we think we got miles (literally) more out of these countries having our own car as opposed to public transportation. We managed to fit in a lot more and it was safe to travel around, even at night, despite the stories we had heard.