As far as I know both of the modules can be used for creating landing pages.
When to use paragraph and when to user panels?
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They are not the same and you can't do the same thing with them.
The Panels module allows a site administrator to create customized layouts for multiple uses.
Paragraphs module comes with a new "paragraphs" field type that works like Entity Reference's.
With Paragraphs you can create collections of items, like you can do with Field collection, as they are fields that you can attach to your content types, you can't do this with Panels.
In the other hand if you want to modify your layout in a simple and easy way you can use Panels, and if you have a collection of items maybe you have used Paragraphs in the content that you're showing.