In practice, they are very similar, but a Processor is more limited than a Bean. I generally use a Processor for simple use cases that just interact with the Exchange. Also, inline processors are a great way to interact without having to create a separate class.
Beans provide more flexibility and also support a true POJO approach. This allows you to more easily integrate with existing APIs (just need to convert the inputs/outputs to match, etc).
Beans also provide great features/flexibility with regards to Camel routing/EIP integration, including…
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rich set of bindings that allow you to quickly bind data from the Exchange to attributes of a bean method, etc.
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POJO consuming/producing allow you to interact with endpoints in a reusable manner
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used as expressions/predicates (for POJO EIP implementation…filters, etc)