Historically, "limelight" referred to a type of stage lighting created by heating a cylinder of lime with an oxyhydrogen flame, which produced a bright, focused beam. That’s where the phrase to be in the limelight comes from—it originally meant being at the center of attention on stage. Today, it’s used more broadly to mean being in the public eye.
To be in the spotlight is a more modern version with the same meaning. Both expressions are still commonly used, and neither one has a particularly positive or negative tone—they’re generally neutral.