I'm not sure I see any reseblance between The Forbidden Planet and Star Trek other than they are both sci-fi and both had similer visual styles. (At least if you are talking about TOS, which I assume you are.) There's not much connection in the story. Forbidden planet is also not an original story. It's a sci-fi telling of Shakespears "The Tempest". Of course Shakespear probably stole it from some Italian folk tale or something.
Here's a website that shows the connection between the two stories.
LINK