We use water to cool down some machines because water has high heat capacity, which means it takes more energy to increase 1 degree. Heat transfer occurs only when temperature is different. We cool down the machine because we don't want it to be too hot, but what does the hot mean? The temp or the heat? When we touch water and another liquid at the same temperature but contain different heat. We feel the heat or the temp?
Regarding the last part "Do we feel heat or the temp?" - this is more physiological then chemical, but I want to share this anyway.
We feel heat flux, i.e. how fast heat is removed or supplied to our skin. Since heat flux can be both advection in the surrounding medium (e.g. air) or conduction in the skin it depends on the situation which is dominant.
In general you can write this as $$ Q=k A \Delta T$$ where $Q$ is the heat flux, $k$ is some transfer coefficient, $A$ is the exchange area and $\Delta T$ is the temperature difference. The value of $k$ depends on the mechanism of transfer e.g. for advection it scales with $\rho C_p v$ i.e. it depends on the heat capacity and flow velocity of the surrounding medium.
The consequence of feeling heat flux is that we can only feel temperature differences not absolute temperature and that advection gives rise to a phenomenon called 'wind chill' that causes us to feel colder when there is more wind, because the higher wind velocity increases the heat transfer coefficient.
Sensation of heat via the nervous system is complicated, as there are different kinds of proteins which respond to temperature and take part in thermoception. The proteins themselves respond directly to temperature, but we don't sense the temperature-based conformation directly. Rather the impact of this change on ion flux leads to a change in electrical potential across the cellular membrane, ultimately resulting in the 'firing' of neural impulses. As such, higher heat transfer results in the sensation of more heat due to more rapid change in membrane potential even though the proteins are directly responding to absolute temperature.
This is further complicated by inhibitory neurons in the sensory processing system that 'accommodate' for ongoing signals by down-regulating a sensation when it occurs continuously. This is why there is a qualitative difference between the feeling of initially stepping out of air conditioning into a hot day and how it feels after being out in the heat for a while.
Note that the feedback from the various sensors are further integrated to provide more complex sensory signals. For instance, the sensation of intense (painful) heat can actually be created by suppressing the sensation of intense cold using cool stimuli. This is the basis for the thermal grid illusion where contact with alternating warm and cool bars can create the illusion of painful heat.
In summary, our sensation of hot and cold is not a simple direct relationship to heat or heat transfer, though it certainly correlates in a useful way that helps us to avoid damage to our bodies under 'typical' conditions.