What is the difference? The difference is based on how the heat is stored.
In an in-floor system the thermal mass can store heat energy like a battery. As the room loses heat through the walls and windows the floor radiates off an equal amount of energy to compensate for the loss. Once this energy is used the temperature will drop, causing the thermostat to turn on and fire the boiler system. Placing the heating elements or tubing under the surface of the floor is normally how this type of heating is constructed. A wet surface such as concrete can be poured over top, or a dry surface such as flooring can be placed on top of the heating element or tubing.
Under-Floor Radiant Heat
With an under-floor system, there is very little heat energy stored. The tubing is installed against the bottom of the floor joists and transfers the heat energy through the plywood using mainly conduction and convection heat transfer. The plywood holds very little heat energy. As the room loses heat energy the floor has no extra to radiate so the thermostat turns on more frequently.
Modern floor heating systems use either electrical resistance elements (electric systems) or fluid flowing in pipes (hydronic systems) to heat the floor. Either type can be installed as the primary, whole-building heating system or as localized floor heating for thermal comfort. Electrical resistance can only be used for heating but when space cooling is also required, hydronic systems must be used.
Other applications that electric or hydronic systems are suited for include snow/ice melting for sidewalks, driveways and landing pads, turf conditioning of football and soccer fields and frost prevention in freezers and skating rinks.
Electric heating elements or hydronic piping can be placed in a concrete floor slab. They can also be placed under the floor covering or attached directly to a wood sub floor.
Some commercial buildings are designed to take advantage of thermal mass that is heated or cooled during off peak hours when utility rates are lower. With the heating/cooling system turned off during the day, the concrete mass and room temperature drift up or down within the desired comfort range. These systems are known as thermally activated building systems or TABS.