The Earth would be so much colder nothing could survive
Using apparent magnitude as a rough guide:
Example: Sun and Moon
What is the ratio in brightness between the Sun and the full Moon?
The apparent magnitude of the Sun is −26.74 (brighter), and the mean magnitude of the full moon is −12.74 (dimmer).
Difference in magnitude:
$x=m_{1}-m_{2}=(-12.74)-(-26.74)=14.00.$
Brightness factor:
$v_{b}=10^{0.4x}=10^{0.4\times 14.00}\approx 398\,107.$
The Sun appears about 400,000 times brighter than the full moon
So if the Sun is now transfering 400,000 times less light it also transfers 400,000 times less heat.
From here we see:
The bottom line is that, of the total heat reaching the surface of the Earth of (1.8+0.0000058) = 1.8000058 watts/cm^2, only 0.0000058/1.8 = 0.0003% is contributed by the Earth's internal heat. This, of course, will dominate everything else if the Sun were to magically vanish!
So of the heat reaching the Earth almost 100% of this is from the sun. Lets take an average of $14^{o}C$ or $287^{o}K$ - dividing this by 400,000 (less than $1^{o}K$) and its so cold even the hardy Tardigrade won't survive past a few minutes.