power consumption of chips at idle

Solution 1:

There will not be a significant difference in idle power usage - might even be identical. Get the „real“ CPU. The low TDP variants are for use cases when your cooling capabilities are severely constrained.

Solution 2:

The two processors have the same Haswell microarchitecture and idle at essentially the same power level. The low-power processor is there for server workloads that demand the lowest possible power consumption and do not require high performance. A typical use case for the E3-1220L v3 would be a ultra-compact, low-power NAS or web server where very small size, minimal fan noise, and low energy usage are required.

In your case, it's unlikely you're that constrained in terms of power usage or cooling. Go with the faster processor.