How precisely can we sense temperature differences?
Short answer
Temperature differences of 0.02 degrees Celcius can be distinguished, dependent on various factors including experimental conditions and bodily location.
Background
The ability to discriminate temperature differences depends on whether it is a cooling or heating pulse, the skin temperature, the duration of the temperature stimulus, age, bodily location among other factors. Unfortunately I cannot access the primary literature other than a few isolated smaller studies. However, Scholarpedia has a very nice entry and associated references, and I quote:

The thermal sensory system is extremely sensitive to very small changes in temperature and on the hairless skin at the base of the thumb, people can perceive a difference of 0.02-0.07 °C in the amplitudes of two cooling pulses or 0.03-0.09 °C of two warming pulses delivered to the hand. The threshold for detecting a change in skin temperature is larger than the threshold for discriminating between two cooling or warming pulses delivered to the skin. When the skin at the base of the thumb is at 33 °C, the threshold for detecting an increase in temperature is 0.20 °C and is 0.11 °C for detecting a decrease in temperature.
The rate that skin temperature changes influences how readily people can detect the change in temperature. If the temperature changes very slowly, for example at a rate of less than 0.5 °C per minute, then a person can be unaware of a 4-5 °C change in temperature, provided that the temperature of the skin remains within the neutral thermal region of 30-36 °C. If the temperature changes more rapidly, such as at 0.1 °C/s, then small decreases and increases in skin temperature are detected. [...]

You also were interested in the palmar skin: here average difference limens are 0.02 - 0.06 °C, dependent on the temperature step used. These values were obtained using cooling pulses. In other words, differences in two cooling pulses of 0.02 - 0.06 °C were discernible, with higher limens needed for higher pulses. Baseline temperatures (29 - 39 °C) had little effect (Darian-Smith et al., 1977).
Reference
Darian-Smith et al., J Invest Dermat 1977;69:146-53
Note
The Stevens, J. C., &amp; Choo, K. K. (1998). Temperature sensitivity of the body surface over the life span. Somatosensory &amp; motor research, 15(1), 13-28 article as commented on by Corvus is indeed a must-read, but my uni library does not have access to it unfortunately.