Casting all your care upon him; for he careth for you. The two halves of the verse use different words, and the difference is the point. The first care is anxiety, the churning kind. The second is concern — the way you care about something because it matters to you. Cast your anxiety on him, because you matter to him.
Peter attaches this to humility. The sentence before is humble yourselves therefore under the mighty hand of God, and the casting is grammatically part of that humbling. Which is unexpected — we usually think of worry as a private struggle, not a pride problem. But anxiety often is a refusal to be a creature: an insistence on holding what we cannot control because handing it over would mean admitting we were never the one holding it.
The image is deliberately physical. To cast is to throw, and a thing thrown has left your hand. It is not the same as thinking about your worry more carefully in God's general direction. Something is meant to actually change hands.
Peter's readers were being persecuted; his letter is not written to people with small troubles. He is not minimizing. He is telling them where to put the weight — with a God who is not merely capable of holding it but personally interested in the one who is tired of holding it.