Do humans perceive temperature or heat-flux? (or both?)
This is the modified answer in response to the discussion:

Facts:


There are warmth and cold receptors in the body at two places: The Peripheral receptors and the Central Receptors
The peripheral receptors are present in skin and the central receptors in the body core at multiple sites the notable site being the hypothalamus
The Temperature receptors have two patterns of firing - The tonic response (sustained firing rate due to steady rise and maintainance in temperature over long time) and the phasic response (also called dynamic response - response to short changes in the temperature)
The Warmth receptors fire maximally at 44 - 46 degrees celcius. The firing stops almost completely at about 48-49 degrees celcius beyond which there is no effective thermoregulation
The Cold receptors fire maximally ar 20 - 35 degrees celcius.




What is measured?


The peripheral receptors sense the temperature of skin.
The central receptors measure the core body temperature.


Only the temperature is measured as WYSIWYG mentions. 

However the frequency of stimulus (How frequently the nerves fire - this depends on the temperature range in which the respective receptors fire) and the two types of receptors present on the skin allows the brain to interpret the flux. So Chris Stronks is correct in telling the heat flux is perceived.

Mechanism of feeling cold:

When you touch a cold object the heat is lost through conduction and radiation through the portion of skin that is in contact with the object. The cold will cause vasoconstriction of that part of skin - This means that the temperature of that part of skin rapidly equilibriates with that of the object. As there is rapid cooling (assuming that the object is cold) the rate of firing of the warmth receptors decrease and the rate of firing of the cold receptors increase. This allows the brain to interpret that the skin is losing heat (the flux).

The radiation part of heat loss is more appreciated when you pass a uncurtained window during the winter - the temperature difference is so great that very high loss of heat occurs through radiation that you shiver. You have to note that during these instances there is no decrease in the core body temperature. The actual decrease in core body heat needs a long exposure to such conditions. In the scenarios detailed above the heat loss is primarily from skin which reflects the ambient conditions (very quickly). Thus the peripheral receptors effect Anticipatory Control which is effectively measuring the flux of heat through skin.

The opposite occurs when you suddenly stand beneath the sun coming from a cold room - the radiant temperature of the sun being so great will cause the skin temperature to rapidly rise causing the body to feel very hot (though the core body temperature has not changed much).

Interesting details that may increase your understanding (though not directly related to the question):

The heat control effected through sensing the change in the skin temperature is called Anticipatory Control. In this the brain "anticipates" what is going to happen - for example a sudden cold draught - and will respond to it - reponds by increasing the heat production (shivering).

The heat control effected through sensing the core body temperature is Negative Feedback Control. In this the brain responds to increase in the core body temperature and takes steps to remedy this (the response will bring the temperature back to the set point).



For more see:Boron &amp; Boulpaep Textbook of Medical Physiology, 2nd Edition, Chapter 59.