I was wondering if there were any difference between these two verbs, especially in their usual respective contexts.
Bonus point for pointing out differences between Argentina and Spain as well (if any!).
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In the same context, both words mean almost the same. The relevant definitions from the RAE are:
Extrañar: 4. tr. Echar de menos a alguien o algo, sentir su falta.
Añorar 1. tr. Recordar con pena la ausencia, privación o pérdida de alguien o algo muy querido.
'Extrañar' is more generic, it just means that you miss something or someone. For example, if I move to UK, I will find driving in the left side of the road a bit weird because I'm used to drive in the right side, so "yo extrañaré conducir por la derecha". Usually 'extrañar' means just a change in your habits, and the new situation will not bother you for too long.
'Añorar' is more about feelings. If I move overseas, I will miss my family, because I love them. It is not a temporary situation, something I will get used to; I will miss them forever.
For completness: 'echar de menos' is an idiom that holds both nuances, and in my opionion, it's more used than 'extrañar' or 'añorar'.
Note: This answer is about Spain. I don't know if they use the words in the same way in Argentina.
Este es el lugar ideal para que me corrijan. Yo si veo una diferencia significativa.
Extrañar siempre se refiere a algo que ya tuvistes. Sin embargo se puede añorar un sueno, un objetivo, ...es un deseo "intimo".
I see a major difference: Extrañar allways bind with something you have had. But añorar could also refer to something you just want to achieve. One can "miss" a dream, a goal, a desire.