They are both versions of the story of love between a princess/consort to the emperor, and the southern prince (Xiao NanChuan Wang).

A Tale of Love and Loyalty is 10-minute episodes adapted from Zhou Fu Yao's novel.
One and Only is 45 minute episodes adapted from MoBao FeiBao's novel.

Similarities
* Slow "slice-of-life" romances
* strong FL and capable supportive ML
* No rushing to get through the plot
* Family and friends important characters

MET is high-school and college ages. MY is stable career.

* Three part drama (ML, ML-with-amnesia, ML-recovers memories)
* In both, FL falls for ML early
* In Be my princess (part 1) ML falls for FL; in Prince-Frog (part 1) ML hates FL
* In both (part 2 and part 3) ML falls for FL
* normal-ML is cold and warms up
* In both (part 3) 2FL lies to ML about FL

Both FL are forthright strong characters and really good at what they do
Both FL are criminologists (coroner and detective) working with ML
Both ML work in important crime-solving positions

I preferred Under the Power, primarily for acting. Imperial Coroner had more interesting cases.