なかなか～ない・あまり～ない: the difference
なかなか～ないN3
Meaning
Not easily, Just won't, Takes a long time to
Past papers
Tested 3 times in past papers
Example
このシステムはなかなか改善されない。
This system just isn't getting any better.
あまり～ないN4
Meaning
Not very, Not much, Not really, Hardly
Past papers
Tested once in past papers
Example
このケーキはあまり美味しくない。
This cake isn't very delicious.
How to tell them apart
「なかなか～ない」 is 'wanted, but it doesn't work out'. 「全然～ない」 is 'not at all': 「全然勉強しない」 (never studies). 「あまり～ない」 is 'not very'.
It can't be used for things that were never going to happen. Don't say 「×冬にはなかなか桜が咲かない」.