新しい計画
新たな計画
Both mean new plan (I think).
Is there any logic behind would you use one over the other? (My brain is defaulting to 新しい as it’s the older word I learnt)
新た is a more literary or formal form, it is mainly used in writing for effect.
I did a bit of research into the etymology you might find interesting.
The word for new was 新たし(あらたし）
For some reason it then became, withthe ら and た becoming switched 新し(あたらし）。
Then it eventually became 新しい as we know it now.
The original form was preserved in the form of 新た as a な adjective.
新しい will be what you will use mostly, I think using 新た in speech would sound weird.
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You can think of 新たな as “brand new”, something somehow significantly different from old version. I feel like there is a qualitiative different when you use 新たな. At least that’s what I think when I see it but I don’t have any proof or reference for you, so it would be better to wait for more responses.
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I think 真新しい is better for brand new? Same grammar point as 真っ赤 and 真っ白い。