Moreover, both agree that there are extenuating circumstances that can change the balance in specific cases. And they agree on the particular extenuating circumstances that are most likely to arise: risk of severe deterioration to woman's physical health, and fetal abnormalities.
All you're pointing out is that Denmark recognizes additional extenuating circumstances for special cases. Specifically, the health exception covers the risk of "severe deterioration of woman's physical or mental health." But note that the "risk to a woman's life or to her physical or mental health should be based solely or principally on circumstances of a medical character." The other grounds for second trimester abortions require unanimous approval from a special committee: https://cyber.harvard.edu/population/abortion/Denmark.abo.ht.... They are not an open-ended exception to allow second trimester abortions in ordinary cases.
I had to look this up and while it's somewhat true, it's highly misleading. "On demand" abortion in Denmark is limited until 12 weeks. However you can still get an abortion afterwards "if the woman's life or health are in danger" or "if certain circumstances are proved to be present (such as poor socioeconomic condition of the woman, risk of birth defects in the baby, the pregnancy being the result of rape, or mental health risk to mother)."[1]
The special circumstances allowed by the Mississippi abortion ban are much more narrow, including only medical emergencies and severe fetal abnormality[2].
[1] https://en.wikipedia.org/wiki/Abortion_in_Denmark
[2] https://www.khou.com/article/news/local/texas/mississippi-ab...