What's the difference between 降参する and 諦める?
Both seems to have the meaning of "to give up", for example:
ぜったい降参しない。(I will never give up.)
ぜったい諦めない。(I will never give up.)
What's the difference in their nuances / meaning?
I think the difference between 降参する and 諦める is the difference between "to surrender" and "to give up".
Giving up is more an emotional state of not having any hope left. Surrendering means that you "put down your arms", officially, but that doesn't necessarily mean you have given up hope.
In that sense, 降参する makes more sense in a formal context of two parties against one another, whereas 諦める can be used in all sorts of situations. E.g.
One does not imply the other. You can surrender as a strategic move, without having given up. You can have given up (hope for life), and choose not to surrender until you die.