What are the differences between hot and cold roof systems?
Cold and hot roofs are an unfortunate terminology that leads to misconceptions. I shouldn't complain, since I'm not sure I have a better idea. Perhaps ventilated or unventilated sheathing? That is the deciding factor.

In cold roofs, the outside air is allowed to freely flow under the roof sheathing. In hot roofs, the insulation is typically installed close to the roof sheathing, but the main characteristic is the space under the sheathing is closed to fresh air flow. Thus, your unventilated roof is indeed a hot roof. You should be aware that according to most building codes, all structural voids not filled with some material should be ventilated.

The main advantage of cold roofs is for cold climates. Snow is much less likely to melt, and if it does melt, ice dams are less likely to form, or at least will be much smaller. With unventilated roofs, heat leaking from the building melts the snow. The water flows down to the eaves, which are much colder, and freezes. This causes a tiny dam which inhibits the water flow, which causes more water to freeze, making a bigger dam. The cycle repeats, and many tons of solid ice can accumulate.

Not only is the weight on an overhang an issue, but the water that hasn't frozen yet just pools. Most sloped roofs are not designed to resist pooling water. It easily backs up under the shingles and other joints and leaks into the building. This is particularly a problem in valleys.

In warm climates, a cold roof is still an advantage, but much less critical. The ventilation prevents any significant temperature differential from building up in the roofing system. The smaller the temperature change, the longer the roof will last. The ventilation will also carry off any moisture that may enter the space, avoiding mold issues. Unventilated roofs must have a good reliable vapor barrier, because any moisture entering the space will have nowhere to go.

The only advantage I can think of for a hot unventilated roof is it allows cheap, easy construction of vaulted ceilings without any special details to ensure the sheathing is being ventilated. These special details often end being a double sheathing system which significantly adds to the cost of a vaulted cold roof system.

You can use foam insulation of any type with either roof system. It's all a matter of correctly detailing the roof system to ensure it is either ventilated or that all voids are closed and filled.