I understand that there is moderate to strong correlation between land surface temperature (LST) and air temperature. However, I am wondering if a given difference in LST is as significant as that of air temperature?
You should be careful with the statement "strong correlation". The LST is a radiative based variable. It has lots to do with radiative forcing on the one end, and material properties of the surface of interest on the other end. The air temperature is an aerodynamic properties. You may find many situations where there is very little or no direct correlation between the two, especially in terms of daily pattern. As an example, if your surface is a city landscape, you'll get a strong variability of LST in a given street while air temperature will be "rather" homogeneous. If you consider a mix of mineral and well irrigated vegetation, you'll find that both temperatures seem disconnected from each other. And in some cases where you get some strong advection, again very little correlation.
The correlation may still be clear, but will depend on space scale, time scale and surface heterogeneity of your area of interest.
Now, regarding the significance of 1K in LST, you should elaborate a bit more. Do you mean in terms of temperature measurement ?
If yes : it's rather common to be able to measure air temperature with a resolution of 1/10 of Celsius degrees with standard meteorological probes, while you hardly measure LST with an absolute uncertainty lower than 2 Celcius degrees (partly because what is measured is brightness temperature, and actual surface emissivity is very difficult to approach).
Depending on your land use heterogeneity it is not much. Vegetated surfaces exposed to higher temperatures (as agricultural fields) are usually 2 to 5 degrees above meteorological air surface. Of course, it depends on surface and atmosphere conditions (wind, moisture, irradiation). At urban environment and city landscape it is much more complicated because, as Mr. Jerome explained very well, heterogeneity at spatial scale and near surface conditions of atmosphere and irradiation increases. In this situation, correlation between them will decrease too.