It is true that dictionary definitions don't always clarify such differences. Take for example Collins:
If a person is accursed, they have been cursed. [literary]
However, ambiguous as it may seem, this short definition already points out two differences:
As @Barmar wrote in a comment, accursed is often used as an adjective (and has survived in such names as The Accursed Mountains, that you mention, or The Accursed Lake, The Accursed City, etc.). Whereas cursed is used as the past participle of the verb to curse (See also this Ngram).
Collins labels accursed as literary, even old fashioned. So if you use it, you show your intention to evoke a past epoch or use a more dramatic register of language.
Also, the same dictionary indicates a usage of accursed to describe things that deserve to be cursed:
Some people use accursed to describe something which they are very annoyed about.
And we need something a little better than this accursed nonsense about the survival of the fittest.
Note that Collins specifies that this meaning is used prenominally:
(prenominal)
hateful; detestable; execrable
But when you describe something as cursed, it means:
experiencing problems and unhappiness (HAVING BAD LUCK)
I think my car is cursed - it never starts when I need it. (Cambridge)